Chapter Questions
The conjugate of $-4-3 i$ is ___________ . (p. A94)
A ________ is a quantity that has both magnitude and direction.
In a triangle with sides $a, b, c$ and angles $A, B, C$, the Law of Cosines states that ___________ .(p. 579)
The distance $d$ from $P_1=\left(x_1, y_1\right)$ to $P_2=\left(x_2, y_2\right)$ is $d=$ ____________. (p. 3)
True or False If $\mathbf{u}$ and $\mathbf{v}$ are parallel vectors, then $\mathbf{u} \times \mathbf{v}=\mathbf{0}$.
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.Plot the point whose rectangular coordinates are $(3,-1)$. What quadrant does the point lie in? (p. 2)
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. If the rectangular coordinates of a point are $(4,-6)$, the point symmetric to it with respect to the origin is_______ (pp. 12-14)
The sum formula for the sine function is $\sin (A+B)=$ ______________. (p. 532)
If $\mathbf{v}$ is a vector, then $\mathbf{v}+(-\mathbf{v})=$ _______.
If $\mathbf{v}=a_1 \mathbf{i}+b_1 \mathbf{j}$ and $\mathbf{w}=a_2 \mathbf{i}+b_2 \mathbf{j}$ are two vectors, then the is ______ ______defined as $\mathbf{v} \cdot \mathbf{w}=a_1 a_2+b_1 b_2$.
In space, points of the form $(x, y, 0)$ lie in a plane called the ________.
True or False For any vector $\mathbf{v}, \mathbf{v} \times \mathbf{v}=\mathbf{0}$.
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. To complete the square of $x^2+6 x$, add
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. The difference formula for $\operatorname{cosine}$ is $\cos (A-B)=$________ (p. 529)
The sum formula for the cosine function is $\cos (A+B)=$ ___________. (p. 529)
A vector $\mathbf{u}$ for which $\|\mathbf{u}\|=1$ is called $a(n)$ _______ vector.
If $\mathbf{v} \cdot \mathbf{w}=0$, then the two vectors $\mathbf{v}$ and $\mathbf{w}$ are ________.
If $\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$ is a vector in space, the scalars $a, b, c$ are called the __________ of $\mathbf{v}$.
True or False If $\mathbf{u}$ and $\mathbf{v}$ are vectors, then $\mathbf{u} \times \mathbf{v}+\mathbf{v} \times \mathbf{u}=\mathbf{0}$.
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.If $P=(a, b)$ is a point on the terminal side of the angle $\theta$ at a distance $r$ from the origin, then $\tan \theta=$ - (p. 425)
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. The standard equation of a circle with center at $(-2,5)$ and radius 3 is_______ (pp. 34-38)
$\sin 120^{\circ}=$ _________$; \cos 240^{\circ}=$ ____________. (pp. 424-431)
If $\mathbf{v}=\langle a, b\rangle$ is an algebraic vector whose initial point is the origin, then $\mathbf{v}$ is called $a(n)$ ________ vector.
If $\mathbf{v}=3 \mathbf{w}$, then the two vectors $\mathbf{v}$ and $\mathbf{w}$ are _________.
The squares of the direction cosines of a vector in space add up to ____________.
True or False $\mathbf{u} \times \mathbf{v}$ is a vector that is parallel to both $\mathbf{u}$ and $\mathbf{v}$.
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. $\tan ^{-1}(-1)=$ .(pp. 498-500)
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Is the sine function even, odd, or neither? (p. 443)
In the complex plane, the $x$-axis is referred to as the ______ axis, and the $y$-axis is called the _____ axis.
If $\mathbf{v}=a \mathbf{i}+b \mathbf{j}$, then $a$ is called the _________ component of $\mathbf{v}$ and $b$ is called the __________ component of $\mathbf{v}$.
True or False Given two nonzero vectors $\mathbf{v}$ and $\mathbf{w}$, it is always possible to decompose $\mathbf{v}$ into two vectors, one parallel to $\mathbf{w}$ and the other orthogonal to $\mathbf{w}$.
True or False In space, the dot product of two vectors is a positive number.
True or False $\|\mathbf{u} \times \mathbf{v}\|=\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta$, where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.
The origin in rectangular coordinates coincides with the______ in polar coordinates; the positive $x$-axis in rectangular coordinates coincides with the___,____ in polar coordinates.
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. $\sin \frac{5 \pi}{4}=$________ . (pp. 424 431)
When a complex number $z$ is written in the polar form $z=r(\cos \theta+i \sin \theta)$, the nonnegative number $r$ is the ______ or ______ of $z$, and the angle $\theta, 0 \leq \theta<2 \pi$, is the ______ of $z$.
If $\mathbf{F}_1$ and $\mathbf{F}_2$ are two forces simultaneously acting on an object, the vector sum $\mathbf{F}_1+\mathbf{F}_2$ is called the ________ force.
True or False Work is a physical example of a vector.
True or False A vector in space may be described by specifying its magnitude and its direction angles.
True or False The area of the parallelogram having $\mathbf{u}$ and $\mathbf{v}$ as adjacent sides is the magnitude of the cross product of $\mathbf{u}$ and $\mathbf{v}$.
If $P$ is a point with polar coordinates $(r, \theta)$, the rectangular coordinates $(x, y)$ of $P$ are given by $x=$ and $y=$
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. $\cos \frac{2 \pi}{3}=$_______ . (pp. 424 431)
Let $z_1=r_1\left(\cos \theta_1+i \sin \theta_1\right)$ and $z_2=r_2\left(\cos \theta_2+i \sin \theta_2\right)$ be two complex numbers. Then$$z_1 z_2=\ldots[\cos (\ldots)+i \sin (\ldots)] .$$
True or False Force is an example of a vector.
The angle $\theta, 0 \leq \theta \leq \pi$, between two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$ can be found using which of the following formulas?(a) $\sin \theta=\frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}$(b) $\cos \theta=\frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}$(c) $\sin \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \mid \mathbf{v} \|}$(d) $\cos \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \mid \mathbf{v} \|}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s). $y=0$
In Problems 7-14, find the value of each determinant. $\left|\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right|$
For the point with polar coordinates $\left(1,-\frac{\pi}{2}\right)$, which of the following best describes the location of the point in a rectangular coordinate system?(a) in quadrant IV(b) on the $y$-axis(c) in quadrant II(d) on the $x$-axis
An equation whose variables are polar coordinates is called $\mathrm{a}(\mathrm{n})$_______,_____
If $z=r(\cos \theta+i \sin \theta)$ is a complex number, then $z^n=\ldots[\cos (\ldots)+i \sin (\ldots)]$.
True or False Mass is an example of a vector.
If two nonzero vectors $\mathbf{v}$ and $\mathbf{w}$ are orthogonal, then the angle between them has which of the following measures?(a) $\pi$(b) $\frac{\pi}{2}$(c) $\frac{3 \pi}{2}$(d) $2 \pi$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s). $x=0$
In Problems 7-14, find the value of each determinant. $\left|\begin{array}{rr}-2 & 5 \\ 2 & -3\end{array}\right|$
The point $\left(5, \frac{\pi}{6}\right)$ can also be represented by which of the following polar coordinates?(a) $\left(5,-\frac{\pi}{6}\right)$(b) $\left(-5, \frac{13 \pi}{6}\right)$(c) $\left(5,-\frac{5 \pi}{6}\right)$(d) $\left(-5, \frac{7 \pi}{6}\right)$
True or False The tests for symmetry in polar coordinates are always conclusive.
Every nonzero complex number will have exactly distinct complex cube roots.
If $\mathbf{v}$ is a vector with initial point $\left(x_1, y_1\right)$ and terminal point $\left(x_2, y_2\right)$, then which of the following is the position vector that equals $\mathbf{v}$ ?(a) $\left\langle x_2-x_1, y_2-y_1\right\rangle$(b) $\left\langle x_1-x_2, y_1-y_2\right\rangle$(c) $\left\langle\frac{x_2-x_1}{2}, \frac{y_2-y_1}{2}\right\rangle$(d) $\left\langle\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right\rangle$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{w} ;(c)$ state whether the vectors are parallel, orthogonal, or neither.$\mathbf{v}=\mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s). $z=2$
In Problems 7-14, find the value of each determinant.$\left|\begin{array}{rr}6 & 5 \\ -2 & -1\end{array}\right|$
True or False In the polar coordinates $(r, \theta), r$ can be negative.
To test whether the graph of a polar equation may be symmetric with respect to the polar axis, replace $\theta$ by__________.
True or False The polar form of a nonzero complex number is unique.
If $\mathbf{v}$ is a nonzero vector with direction angle $\alpha, 0^{\circ} \leq \alpha<360^{\circ}$, between $\mathbf{v}$ and $\mathbf{i}$, then $\mathbf{v}$ equals which of the following?(a) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}-\sin \alpha \mathbf{j})$(b) $\|\mathbf{v}\|(\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j})$(c) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}-\cos \alpha \mathbf{j})$(d) $\|\mathbf{v}\|(\sin \alpha \mathbf{i}+\cos \alpha \mathbf{j})$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s). $y=3$
In Problems 7-14, find the value of each determinant. $\left|\begin{array}{rr}-4 & 0 \\ 5 & 3\end{array}\right|$
True or False The polar coordinates of a point are unique.
To test whether the graph of a polar equation may be symmetric with respect to the line $\theta=\frac{\pi}{2}$, replace $\theta$ by______.
If $z=x+y i$ is a complex number, then $|z|$ equals which of the following?(a) $x^2+y^2$(b) $|x|+|y|$(c) $\sqrt{x^2+y^2}$(d) $\sqrt{|x|+|y|}$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY)$\mathbf{v}+\mathbf{w}$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-2 \mathbf{j}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s). $x=-4$
In Problems 7-14, find the value of each determinant. $\left|\begin{array}{lll}A & B & C \\ 2 & 1 & 4 \\ 1 & 3 & 1\end{array}\right|$
In Problems 11-18, match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(2,-\frac{11 \pi}{6}\right)$
True or False A cardioid passes through the pole.
If $z_1=r_1\left(\cos \theta_1+i \sin \theta_1\right)$ and $z_2=r_2\left(\cos \theta_2+i \sin \theta_2\right)$ are complex numbers, then $\frac{z_1}{z_2}, z_2 \neq 0$, equals which of the following?(a) $\frac{r_1}{r_2}\left[\cos \left(\theta_1-\theta_2\right)+i \sin \left(\theta_1-\theta_2\right)\right]$(b) $\frac{r_1}{r_2}\left[\cos \left(\frac{\theta_1}{\theta_2}\right)+i \sin \left(\frac{\theta_1}{\theta_2}\right)\right]$(c) $\frac{r_1}{r_2}\left[\cos \left(\theta_1+\theta_2\right)-i \sin \left(\theta_1+\theta_2\right)\right]$(d) $\frac{r_1}{r_2}\left[\cos \left(\frac{\theta_1}{\theta_2}\right)-i \sin \left(\frac{\theta_1}{\theta_2}\right)\right]$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $\mathrm{u}+\mathrm{v}$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=2 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}+2 \mathbf{j}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s). $z=-3$
In Problems 7-14, find the value of each determinant. $\left|\begin{array}{lll}A & B & C \\ 0 & 2 & 4 \\ 3 & 1 & 3\end{array}\right|$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(-2,-\frac{\pi}{6}\right)$
Rose curves are characterized by equations of the form $r=a \cos (n \theta)$ or $r=a \sin (n \theta), a \neq 0$. If $n \neq 0$ is even, the rose has_____ petals; if $n \neq \pm 1$ is odd, the rose has petals.
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.$1+i$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $3 \mathrm{v}$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=\sqrt{3} \mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s).$x=1$ and $y=2$
In Problems 7-14, find the value of each determinant.$\left|\begin{array}{rrr}A & B & C \\ -1 & 3 & 5 \\ 5 & 0 & -2\end{array}\right|$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(-2, \frac{\pi}{6}\right)$
For a positive real number $a$, the graph of which of the following polar equations is a circle with radius $a$ and center at $(a, 0)$ in rectangular coordinates?(a) $r=2 a \sin \theta$(b) $r=-2 a \sin \theta$(c) $r=2 a \cos \theta$(d) $r=-2 a \cos \theta$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $-1+i$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $2 w$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$
In Problems 7-14, describe the set of points $(x, y, z)$ defined by the equation(s).$x=3$ and $z=1$
In Problems 7-14, find the value of each determinant. $\left|\begin{array}{rrr}A & B & C \\ 1 & -2 & -3 \\ 0 & 2 & -2\end{array}\right|$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(2, \frac{7 \pi}{6}\right)$
In polar coordinates, the points $(r, \theta)$ and $(-r, \theta)$ are symmetric with respect to which of the following?(a) the polar axis (or $x$-axis)(b) the pole (or origin)(c) the line $\theta=\frac{\pi}{2}$ (or $y$-axis)(d) the line $\theta=\frac{\pi}{4}$ (or $y=x$ )
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.$\sqrt{3}-i$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $v-w$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-8 \mathbf{j}$
In Problems 15-20, find the distance from $P_1$ to $P_2$. $P_1=(0,0,0)$ and $P_2=(4,1,2)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v}$, (c) $\mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$.$$\begin{aligned}& \mathbf{v}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \\& \mathbf{w}=3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}\end{aligned}$$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$.$\begin{aligned} \mathbf{v} & =2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \\ \mathbf{w} & =3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}\end{aligned}$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(2, \frac{5 \pi}{6}\right)$
In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r=4$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $1-\sqrt{3} i$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $u-v$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=3 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=9 \mathbf{i}-12 \mathbf{j}$
In Problems 15-20, find the distance from $P_1$ to $P_2$. $P_1=(0,0,0)$ and $P_2=(1,-2,3)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v}$, (c) $\mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$. $\mathbf{v}=-\mathbf{i}+3 \mathbf{j}+2 \mathbf{k}$$\mathbf{w}=3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$. $\begin{aligned} & \mathbf{v}=-\mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \\ & \mathbf{w}=3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}\end{aligned}$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(-2, \frac{5 \pi}{6}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r=2$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.$-3 i$
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $3 \mathbf{v}+\mathbf{u}-2 \mathbf{w}$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=4 \mathbf{i}, \quad \mathbf{w}=\mathbf{j}$
In Problems 15-20, find the distance from $P_1$ to $P_2$. $P_1=(-1,2,-3)$ and $P_2=(0,-2,1)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$.$\begin{aligned} \mathbf{v} & =\mathbf{i}+\mathbf{j} \\ \mathbf{w} & =2 \mathbf{i}+\mathbf{j}+\mathbf{k}\end{aligned}$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(-2, \frac{7 \pi}{6}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $\theta=\frac{\pi}{3}$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. -2
In Problems 11-18, use the vectors in the figure at the right to graph each of the following vectors.(GRAPH CANT COPY) $2 u-3 \mathbf{v}+\mathbf{w}$
In Problems 9-18, (a) find the dot product $\mathbf{v} \cdot \mathbf{w} ;(b)$ find the angle between $\mathbf{v}$ and $\mathbf{v}=\mathbf{i}, \quad \mathbf{w}=-3 \mathbf{j}$
In Problems 15-20, find the distance from $P_1$ to $P_2$. $P_1=(-2,2,3)$ and $P_2=(4,0,-3)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$. $\begin{aligned} \mathbf{v} & =\mathbf{i}-4 \mathbf{j}+2 \mathbf{k} \\ \mathbf{w} & =3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}\end{aligned}$
Match each point in polar coordinates with either $A, B, C$, or $D$ on the graph.$\left(2, \frac{11 \pi}{6}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.$\theta=-\frac{\pi}{4}$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $4-4 i$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY)$\mathbf{A}+\mathbf{B}=\mathbf{F}$
Find $a$ so that the vectors $\mathbf{v}=\mathbf{i}-a \mathbf{j}$
In Problems 15-20, find the distance from $P_1$ to $P_2$. $P_1=(4,-2,-2)$ and $P_2=(3,2,1)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$. $\begin{aligned} \mathbf{v} & =2 \mathbf{i}-\mathbf{j}+2 \mathbf{k} \\ \mathbf{w} & =\mathbf{j}-\mathbf{k}\end{aligned}$
In Problems 19-32, plot each point given in polar coordinates.$\left(3,90^{\circ}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r \sin \theta=4$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.$9 \sqrt{3}+9 i$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{K}+\mathbf{G}=\mathbf{F}$
Find $b$ so that the vectors $\mathbf{v}=\mathbf{i}+\mathbf{j}$ and $\mathbf{w}=2 \mathbf{i}+3 \mathbf{j}$ are orthogonal. and $\mathbf{w}=\mathbf{i}+b \mathbf{j}$ are orthogonal.
In Problems 15-20, find the distance from $P_1$ to $P_2$. $P_1=(2,-3,-3)$ and $P_2=(4,1,-1)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$. $\begin{aligned} \mathbf{v} & =3 \mathbf{i}+\mathbf{j}+3 \mathbf{k} \\ \mathbf{w} & =\mathbf{i}-\mathbf{k}\end{aligned}$
Plot each point given in polar coordinates.$\left(4,270^{\circ}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.$r \cos \theta=4$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $3-4 i$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{C}=\mathbf{D}-\mathbf{E}+\mathbf{F}$
In Problems 21-26, decompose $\mathbf{v}$ into two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, where $\mathbf{v}_1$ is parallel to $\mathbf{w}$, and $\mathbf{v}_2$ is orthogonal to $\mathbf{w}$.$\mathbf{v}=2 \mathbf{i}-3 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$
In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. $(0,0,0) ;(2,1,3)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$. $\begin{aligned} \mathbf{v} & =\mathbf{i}-\mathbf{j}-\mathbf{k} \\ \mathbf{w} & =4 \mathbf{i}-3 \mathbf{k}\end{aligned}$
Plot each point given in polar coordinates. $(-2,0)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r \cos \theta=-2$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $2+\sqrt{3} i$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{G}+\mathbf{H}+\mathbf{E}=\mathbf{D}$
In Problems 21-26, decompose $\mathbf{v}$ into two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, where $\mathbf{v}_1$ is parallel to $\mathbf{w}$, and $\mathbf{v}_2$ is orthogonal to $\mathbf{w}$. $\mathbf{v}=-3 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=2 \mathbf{i}+\mathbf{j}$
In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box.$(0,0,0)$;$(4,2,2)$
In Problems 15-22, find (a) $\mathbf{v} \times \mathbf{w},($ b) $\mathbf{w} \times \mathbf{v},(c) \mathbf{w} \times \mathbf{w}$, and (d) $\mathbf{v} \times \mathbf{v}$.$\begin{aligned} \mathbf{v} & =2 \mathbf{i}-3 \mathbf{j} \\ \mathbf{w} & =3 \mathbf{j}-2 \mathbf{k}\end{aligned}$
Plot each point given in polar coordinates. $(-3, \pi)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.$r \sin \theta=-2$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $-2+3 i$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{E}+\mathbf{D}=\mathbf{G}+\mathbf{H}$
In Problems 21-26, decompose $\mathbf{v}$ into two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, where $\mathbf{v}_1$ is parallel to $\mathbf{w}$, and $\mathbf{v}_2$ is orthogonal to $\mathbf{w}$. $\mathbf{v}=\mathbf{i}-\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}-2 \mathbf{j}$
In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. $(1,2,3) ;(3,4,5)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$$u \times v$
Plot each point given in polar coordinates.$\left(6, \frac{\pi}{6}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r=2 \cos \theta$
In Problems 13-24, plot each complex number in the complex plane and write it in polar form. Express the argument in degrees. $\sqrt{5}-i$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{H}-\mathbf{C}=\mathbf{G}-\mathbf{F}$
In Problems 21-26, decompose $\mathbf{v}$ into two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, where $\mathbf{v}_1$ is parallel to $\mathbf{w}$, and $\mathbf{v}_2$ is orthogonal to $\mathbf{w}$. $\mathbf{v}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-2 \mathbf{j}$
In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. $(5,6,1) ;(3,8,2)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $v \times w$
Plot each point given in polar coordinates. $\left(5, \frac{5 \pi}{3}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.$r=2 \sin \theta$
In Problems 25-34, write each complex number in rectangular form.$2\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{A}+\mathbf{B}+\mathbf{K}+\mathbf{G}=\mathbf{0}$
In Problems 21-26, decompose $\mathbf{v}$ into two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, where $\mathbf{v}_1$ is parallel to $\mathbf{w}$, and $\mathbf{v}_2$ is orthogonal to $\mathbf{w}$.$\mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}-\mathbf{j}$
In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. $(-1,0,2) ;(4,2,5)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$$\mathrm{v} \times \mathrm{u}$
Plot each point given in polar coordinates.$\left(-2,135^{\circ}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r=-4 \sin \theta$
In Problems 25-34, write each complex number in rectangular form. $3\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)$
In Problems 19-26, use the figure at the right. Determine whether each statement given is true or false.(GRAPH CANT COPY) $\mathbf{A}+\mathbf{B}+\mathbf{C}+\mathbf{H}+\mathbf{G}=\mathbf{0}$
In Problems 21-26, decompose $\mathbf{v}$ into two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$, where $\mathbf{v}_1$ is parallel to $\mathbf{w}$, and $\mathbf{v}_2$ is orthogonal to $\mathbf{w}$.$\mathbf{v}=\mathbf{i}-3 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$
In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. $(-2,-3,0) ; \quad(-6,7,1)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $w \times v$
Plot each point given in polar coordinates.$\left(-3,120^{\circ}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r=-4 \cos \theta$
In Problems 25-34, write each complex number in rectangular form. $4\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)$
If $\|\mathbf{v}\|=4$, what is $\|3 \mathbf{v}\|$ ?
Given vectors $\mathbf{u}=\mathbf{i}+5 \mathbf{j}$ and $\mathbf{v}=4 \mathbf{i}+y \mathbf{j}$, find $y$ so that the angle between the vectors is $60^{\circ}$. $^{\dagger}$
In Problems $27-32$, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$; that is, find its position vector.$P=(0,0,0) ; \quad Q=(3,4,-1)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $\mathbf{v} \times \mathbf{v}$
Plot each point given in polar coordinates.$\left(4,-\frac{2 \pi}{3}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r \sec \theta=4$
In Problems 25-34, write each complex number in rectangular form. $2\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$
If $\|\mathbf{v}\|=2$, what is $\|-4 \mathbf{v}\|$ ?
Given vectors $\mathbf{u}=x \mathbf{i}+2 \mathbf{j}$ and $\mathbf{v}=7 \mathbf{i}-3 \mathbf{j}$, find $x$ so that the angle between the vectors is $30^{\circ}$.
In Problems $27-32$, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$; that is, find its position vector. $P=(0,0,0) ; \quad Q=(-3,-5,4)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $w \times w$
Plot each point given in polar coordinates.$\left(2,-\frac{5 \pi}{4}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r \csc \theta=8$
In Problems 25-34, write each complex number in rectangular form.$3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector. $P=(0,0) ; \quad Q=(3,4)$
Computing Work Find the work done by a force of 3 pounds acting in the direction $60^{\circ}$ to the horizontal in moving an object 6 feet from $(0,0)$ to $(6,0)$.
In Problems $27-32$, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$; that is, find its position vector. $P=(3,2,-1) ; \quad Q=(5,6,0)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $(3 u) \times v$
Plot each point given in polar coordinates.$\left(-1,-\frac{\pi}{3}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.$r \csc \theta=-2$
In Problems 25-34, write each complex number in rectangular form. $4\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector. $P=(0,0) ; \quad Q=(-3,-5)$
Computing Work A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of $30^{\circ}$ with the horizontal. How much work is done in moving the wagon 100 feet?
In Problems $27-32$, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$; that is, find its position vector. $P=(-3,2,0) ; \quad Q=(6,5,-1)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $\mathbf{v} \times(4 \mathbf{w})$
Plot each point given in polar coordinates.$\left(-3,-\frac{3 \pi}{4}\right)$
Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $r \sec \theta=-4$
In Problems 25-34, write each complex number in rectangular form. $0.2\left(\cos 100^{\circ}+i \sin 100^{\circ}\right)$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector. $P=(3,2) ; \quad Q=(5,6)$
Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun's rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun's rays. Let the vector $\mathbf{A}$ represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure. The total number of watts collected by the panel is given by $W=|\mathbf{I} \cdot \mathbf{A}|$.(IMAGE CANT COPY)Suppose that $\mathbf{I}=\langle-0.02,-0.01\rangle$ and $\mathbf{A}=\langle 300,400\rangle$.(a) Find $\|\mathbf{I}\|$ and $\|\mathbf{A}\|$, and interpret the meaning of each.(b) Compute $W$ and interpret its meaning.(c) If the solar panel is to collect the maximum number of watts, what must be true about $\mathbf{I}$ and $\mathbf{A}$ ?
In Problems $27-32$, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$; that is, find its position vector. $P=(-2,-1,4) ; \quad Q=(6,-2,4)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$$\mathbf{u} \times(2 \mathbf{v})$
Plot each point given in polar coordinates.$(-2,-\pi)$
In Problems 31-38, match each of the graphs $(A)$ through $(H)$ to one of the following polar equation. $r=2$
In Problems 25-34, write each complex number in rectangular form. $0.4\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector. $P=(-3,2) ; \quad Q=(6,5)$
Rainfall Measurement Let the vector $\mathbf{R}$ represent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector A represent the area, in square inches, whose direction is the orientation of the opening of the rain gauge. See the figure. The volume of rain collected in the gauge, in cubic inches, is given by $V=|\mathbf{R} \cdot \mathbf{A}|$, even when the rain falls in a slanted direction or the gauge is not perfectly vertical.Suppose that $\mathbf{R}=\langle 0.75,-1.75\rangle$ and $\mathbf{A}=\langle 0.3,1\rangle$.(a) Find $\|\mathbf{R}\|$ and $\|\mathbf{A}\|$, and interpret the meaning of each.(b) Compute $V$ and interpret its meaning.(c) If the gauge is to collect the maximum volume of rain, what must be true about $\mathbf{R}$ and $\mathbf{A}$ ?(IMAGE CANT COPY)
In Problems $27-32$, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$; that is, find its position vector.$P=(-1,4,-2) ; \quad Q=(6,2,2)$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$$(-3 \mathbf{v}) \times \mathbf{w}$
Plot each point given in polar coordinates.$\left(-3,-\frac{\pi}{2}\right)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation. $\theta=\frac{\pi}{4}$
In Problems 25-34, write each complex number in rectangular form. $2\left(\cos \frac{\pi}{18}+i \sin \frac{\pi}{18}\right)$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector. $P=(-2,-1) ; \quad Q=(6,-2)$
Braking Load A Toyota Sienna with a gross weight of 5300 pounds is parked on a street with an $8^{\circ}$ grade. See the figure. Find the magnitude of the force required to keep the Sienna from rolling down the hill. What is the magnitude of the force perpendicular to the hill?(IMAGE CANT COPY)
In Problems 33-38, find $\|\mathbf{v}\|$. $\mathbf{v}=3 \mathbf{i}-6 \mathbf{j}-2 \mathbf{k}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $\mathbf{u} \cdot(\mathbf{u} \times \mathbf{v})$
In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:(a) $r>0, \quad-2 \pi \leq \theta<0$(b) $r<0, \quad 0 \leq \theta<2 \pi$(c) $r>0,2 \pi \leq \theta<4 \pi$$\left(5, \frac{2 \pi}{3}\right)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation. $r=2 \cos \theta$
In Problems 25-34, write each complex number in rectangular form. $3\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector. $P=(-1,4) ; \quad Q=(6,2)$
Braking Load A Chevrolet Silverado with a gross weight of 4500 pounds is parked on a street with a $10^{\circ}$ grade. Find the magnitude of the force required to keep the Silverado from rolling down the hill. What is the magnitude of the force perpendicular to the hill?
In Problems 33-38, find $\|\mathbf{v}\|$. $\mathbf{v}=-6 \mathbf{i}+12 \mathbf{j}+4 \mathbf{k}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$$\mathbf{v} \cdot(\mathbf{v} \times \mathbf{w})$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which: $\left(4, \frac{3 \pi}{4}\right)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation.$r \cos \theta=2$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.$$\begin{aligned}& z=2\left(\cos 40^{\circ}+i \sin 40^{\circ}\right) \\& w=4\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\end{aligned}$$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector.$P=(1,0) ; \quad Q=(0,1)$
Ramp Angle Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is $20^{\circ}$, how many pounds of force must Timmy exert to hold the piano in position?(IMAGE CANT COPY)
In Problems 33-38, find $\|\mathbf{v}\|$. $\mathbf{v}=\mathbf{i}-\mathbf{j}+\mathbf{k}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$$\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which: $(-2,3 \pi)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation.$r=1+\cos \theta$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.$$\begin{aligned}& z=\cos 120^{\circ}+i \sin 120^{\circ} \\& w=\cos 100^{\circ}+i \sin 100^{\circ}\end{aligned}$$
In Problems 29-36, the vector $\mathbf{v}$ has initial point $P$ and terminal point $Q$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j} ;$ that is, find its position vector.$P=(1,1) ; \quad Q=(2,2)$
Incline Angle A bulldozer exerts 1000 pounds of force to prevent a 5000-pound boulder from rolling down a hill. Determine the angle of inclination of the hill.
In Problems 33-38, find $\|\mathbf{v}\|$. $\mathbf{v}=-\mathbf{i}-\mathbf{j}+\mathbf{k}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $(u \times v) \cdot w$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:$(-3,4 \pi)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation.$r=2 \sin \theta$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.$$\begin{aligned}& z=3\left(\cos 130^{\circ}+i \sin 130^{\circ}\right) \\& w=4\left(\cos 270^{\circ}+i \sin 270^{\circ}\right)\end{aligned}$$
In Problems 37-42, find $\|\mathbf{v}\|$. $\mathbf{v}=3 \mathbf{i}-4 \mathbf{j}$
Find the acute angle that a constant unit force vector makes with the positive $x$-axis if the work done by the force in moving a particle from $(0,0)$ to $(4,0)$ equals 2 .
In Problems 33-38, find $\|\mathbf{v}\|$. $\mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $\mathbf{v} \cdot(\mathbf{u} \times \mathbf{w})$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:$\left(1, \frac{\pi}{2}\right)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation. $\theta=\frac{3 \pi}{4}$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.$$\begin{aligned}& z=2\left(\cos 80^{\circ}+i \sin 80^{\circ}\right) \\& w=6\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)\end{aligned}$$
In Problems 37-42, find $\|\mathbf{v}\|$.$\mathbf{v}=-5 \mathbf{i}+12 \mathbf{j}$
Prove the distributive property:$$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$
In Problems 33-38, find $\|\mathbf{v}\|$.$\mathbf{v}=6 \mathbf{i}+2 \mathbf{j}-2 \mathbf{k}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $(\mathbf{v} \times \mathbf{u}) \cdot \mathbf{w}$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:$(2, \pi)$
Match each of the graphs $(A)$ through $(H)$ to one of the following polar equation.$r \sin \theta=2$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form. $z=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)$
In Problems 37-42, find $\|\mathbf{v}\|$. $\mathrm{v}=\mathrm{i}-\mathrm{j}$
Prove property (5): $\mathbf{0} \cdot \mathbf{v}=0$.
In Problems 39-44, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$. $2 \mathbf{v}+3 \mathbf{w}$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $\mathbf{u} \times(\mathbf{v} \times \mathbf{v})$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:$\left(-3,-\frac{\pi}{4}\right)$
In Problems 39-62, identify and graph each polar equation. $r=2+2 \cos \theta$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form. $z=4\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right)$$w=2\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)$$w=2\left(\cos \frac{9 \pi}{16}+i \sin \frac{9 \pi}{16}\right)$
In Problems 37-42, find $\|\mathbf{v}\|$.$\mathbf{v}=-\mathbf{i}-\mathbf{j}$
If $\mathbf{v}$ is a unit vector and the angle between $\mathbf{v}$ and $\mathbf{i}$ is $\alpha$, show that $\mathbf{v}=\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j}$.
In Problems 39-44, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$.$3 v-2 w$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ $(w \times w) \times v$
Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:$\left(-2,-\frac{2 \pi}{3}\right)$
Identify and graph each polar equation. $r=1+\sin \theta$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.$$\begin{aligned}& z=2+2 i \\& w=\sqrt{3}-i\end{aligned}$$
In Problems 37-42, find $\|\mathbf{v}\|$.$\mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}$
Suppose that $\mathbf{v}$ and $\mathbf{w}$ are unit vectors. If the angle between $\mathbf{v}$ and $\mathbf{i}$ is $\alpha$ and the angle between $\mathbf{w}$ and $\mathbf{i}$ is $\beta$, use the idea of the dot product $\mathbf{v} \cdot \mathbf{w}$ to prove that$$\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$$
In Problems 39-44, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$. $\|\mathbf{v}-\mathbf{w}\|$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$Find a vector orthogonal to both $\mathbf{u}$ and $\mathbf{v}$.
In Problems 41-56, polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(3, \frac{\pi}{2}\right)$
Identify and graph each polar equation. $r=3-3 \sin \theta$
In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.$$\begin{aligned}& z=1-i \\& w=1-\sqrt{3} i\end{aligned}$$
In Problems 37-42, find $\|\mathbf{v}\|$. $\mathbf{v}=6 \mathbf{i}+2 \mathbf{j}$
Show that the projection of $\mathbf{v}$ onto $\mathbf{i}$ is $(\mathbf{v} \cdot \mathbf{i}) \mathbf{i}$. Then show that we can always write a vector $\mathbf{v}$ as$$\mathbf{v}=(\mathbf{v} \cdot \mathbf{i}) \mathbf{i}+(\mathbf{v} \cdot \mathbf{j}) \mathbf{j}$$
In Problems 39-44, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$. $\|\mathbf{v}+\mathbf{w}\|$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$ Find a vector orthogonal to both $\mathbf{u}$ and $\mathbf{w}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(4, \frac{3 \pi}{2}\right)$
Identify and graph each polar equation. $r=2-2 \cos \theta$
In Problems 43-54, write each expression in the standard form $a+b i$.$\left[4\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)\right]^3$
In Problems 43-48, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}$.$2 \mathbf{v}+3 \mathbf{w}$
(a) If $\mathbf{u}$ and $\mathbf{v}$ have the same magnitude, show that $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-\mathbf{v}$ are orthogonal.(b) Use this to prove that an angle inscribed in a semicircle is a right angle (see the figure).(IMAGE CANT COPY)
In Problems 39-44, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$. $\|\mathbf{v}\|-\|\mathbf{w}\|$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$Find a vector orthogonal to both $\mathbf{u}$ and $\mathbf{i}+\mathbf{j}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$(-2,0)$
Identify and graph each polar equation.$r=2+\sin \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $\left[3\left(\cos 80^{\circ}+i \sin 80^{\circ}\right)\right]^3$
In Problems 43-48, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}$.$3 v-2 w$
Let $\mathbf{v}$ and $\mathbf{w}$ denote two nonzero vectors. Show that the vector $\mathbf{v}-\alpha \mathbf{w}$ is orthogonal to $\mathbf{w}$ if $\alpha=\frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^2}$.
In Problems 39-44, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$. $\|\mathbf{v}\|+\|\mathbf{w}\|$
In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.$$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}$$Find a vector orthogonal to both $\mathbf{u}$ and $\mathbf{j}+\mathbf{k}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$(-3, \pi)$
Identify and graph each polar equation. $r=2-\cos \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $\left[2\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)\right]^5$
In Problems 43-48, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}$.$\|v-w\|$
Let $\mathbf{v}$ and $\mathbf{w}$ denote two nonzero vectors. Show that the vectors $\|\mathbf{w}\| \mathbf{v}+\|\mathbf{v}\| \mathbf{w}$ and $\|\mathbf{w}\| \mathbf{v}-\|\mathbf{v}\| \mathbf{w}$ are orthogonal.
In Problems 45-50, find the unit vector in the same direction as $\mathbf{v}$.$v=5 i$
In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$.$P_1=(0,0,0), \quad P_2=(1,2,3), \quad P_3=(-2,3,0)$
Polar coordinates of a point are given. Find the rectangular coordinates of each point. $\left(6,150^{\circ}\right)$
Identify and graph each polar equation. $r=4-2 \cos \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $\left[\sqrt{2}\left(\cos \frac{5 \pi}{16}+i \sin \frac{5 \pi}{16}\right)\right]^4$
In Problems 43-48, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}$. $\|\mathbf{v}+\mathbf{w}\|$
In the definition of work given in this section, what is the work done if $\mathbf{F}$ is orthogonal to $\overrightarrow{A B}$ ?
In Problems 45-50, find the unit vector in the same direction as $\mathbf{v}$. $v=-3 \mathbf{j}$
In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$.$P_1=(0,0,0), \quad P_2=(2,3,1), \quad P_3=(-2,4,1)$
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(5,300^{\circ}\right)$
Identify and graph each polar equation. $r=4+2 \sin \theta$
In Problems 43-54, write each expression in the standard form $a+b i$.$\left[\sqrt{3}\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^6$
In Problems 43-48, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}$. $\|\mathbf{v}\|-\|\mathbf{w}\|$
. Prove the polarization identity,$$\|\mathbf{u}+\mathbf{v}\|^2-\|\mathbf{u}-\mathbf{v}\|^2=4(\mathbf{u} \cdot \mathbf{v})$$
In Problems 45-50, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=3 \mathbf{i}-6 \mathbf{j}-2 \mathbf{k}$
In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$. $P_1=(1,2,0), \quad P_2=(-2,3,4), \quad P_3=(0,-2,3)$
Polar coordinates of a point are given. Find the rectangular coordinates of each point. $\left(-2, \frac{3 \pi}{4}\right)$
Identify and graph each polar equation. $r=1+2 \sin \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $\left[\frac{1}{2}\left(\cos 72^{\circ}+i \sin 72^{\circ}\right)\right]^5$
In Problems 43-48, find each quantity if $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}$. $\|\mathbf{v}\|+\|\mathbf{w}\|$
Create an application (different from any found in the text) that requires a dot product.
In Problems 45-50, find the unit vector in the same direction as $\mathbf{v}$.$\mathbf{v}=-6 \mathbf{i}+12 \mathbf{j}+4 \mathbf{k}$
In Problems 45-48, find the area of the parallelogram with one corner at $P_1$ and adjacent sides $\overrightarrow{P_1 P_2}$ and $\overrightarrow{P_1 P_3}$. $P_1=(-2,0,2), \quad P_2=(2,1,-1), \quad P_3=(2,-1,2)$
Polar coordinates of a point are given. Find the rectangular coordinates of each point. $\left(-2, \frac{2 \pi}{3}\right)$
Identify and graph each polar equation.$r=1-2 \sin \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $\left[\sqrt{5}\left(\cos \frac{3 \pi}{16}+i \sin \frac{3 \pi}{16}\right)\right]^4$
In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$.$v=5 i$
Problems 49-52 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the average rate of change of $f(x)=x^3-5 x^2+27$ from -3 to 2 .
In Problems 45-50, find the unit vector in the same direction as $\mathbf{v}$. $v=i+j+k$
In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$.$$\begin{aligned}& P_1=(1,1,2), \quad P_2=(1,2,3), \quad P_3=(-2,3,0), \\& P_4=(-2,4,1)\end{aligned}$$
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(-1,-\frac{\pi}{3}\right)$
Identify and graph each polar equation. $r=2-3 \cos \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $\left[\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)\right]^6$
In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=-3 \mathbf{j}$
Problems 49-52 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of $5 \cos 60^{\circ}+2 \tan \frac{\pi}{4}$. Do not use a calculator.
In Problems 45-50, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}$
In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$.$\begin{aligned} P_1 & =(2,1,1), \quad P_2=(2,3,1), \quad P_3=(-2,4,1), \\ P_4 & =(-2,6,1)\end{aligned}$
Polar coordinates of a point are given. Find the rectangular coordinates of each point. $\left(-3,-\frac{3 \pi}{4}\right)$
Identify and graph each polar equation. $r=2+4 \cos \theta$
In Problems 43-54, write each expression in the standard form $a+b i$. $(1-i)^5$
In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=3 \mathbf{i}-4 \mathbf{j}$
Problems 49-52 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Establish the identity: $\left(1-\sin ^2 \theta\right)\left(1+\tan ^2 \theta\right)=1$
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$. $\mathbf{v}=\mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+\mathbf{k}$
In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$. $\begin{aligned} P_1 & =(1,2,-1), \quad P_2=(4,2,-3), \quad P_3=(6,-5,2), \\ P_4 & =(9,-5,0)\end{aligned}$
Polar coordinates of a point are given. Find the rectangular coordinates of each point. $\left(-2,-180^{\circ}\right)$
Identify and graph each polar equation.$r=3 \cos (2 \theta)$
In Problems 43-54, write each expression in the standard form $a+b i$. $(\sqrt{3}-i)^6$
In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=-5 \mathbf{i}+12 \mathbf{j}$
Problems 49-52 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Volume of a Box An open-top box is made from a sheet of metal by cutting squares from each corner and folding up the sides. The sheet has a length of 19 inches and a width of 13 inches. If $x$ is the length of one side of the square to be cut out, write a function, $V(x)$, for the volume of the box in terms of $x$.
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$. $\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}-\mathbf{k}$
In Problems 49-52, find the area of the parallelogram with vertices $P_1, P_2, P_3$, and $P_4$. $\begin{aligned} P_1 & =(-1,1,1), \quad P_2=(-1,2,2), \quad P_3=(-3,4,-5) \text {, } \\ P_4 & =(-3,5,-4)\end{aligned}$
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(-3,-90^{\circ}\right)$
Identify and graph each polar equation. $r=2 \sin (3 \theta)$
In Problems 43-54, write each expression in the standard form $a+b i$.$(\sqrt{2}-i)^6$
In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$. $\mathbf{v}=\mathbf{i}-\mathbf{j}$
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$. $\mathbf{v}=2 \mathbf{i}+\mathbf{j}-3 \mathbf{k}, \quad \mathbf{w}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}$
Find a unit vector normal to the plane containing $\mathbf{v}=\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}+\mathbf{j}+3 \mathbf{k}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(7.5,110^{\circ}\right)$
Identify and graph each polar equation.$r=4 \sin (5 \theta)$
In Problems 43-54, write each expression in the standard form $a+b i$. $(1-\sqrt{5} i)^8$
In Problems 49-54, find the unit vector in the same direction as $\mathbf{v}$.$\mathbf{v}=2 \mathbf{i}-\mathbf{j}$
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$. $\mathbf{v}=2 \mathbf{i}+2 \mathbf{j}-\mathbf{k}, \quad \mathbf{w}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$
Find a unit vector normal to the plane containing $\mathbf{v}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$ and $\mathbf{w}=-2 \mathbf{i}-4 \mathbf{j}-3 \mathbf{k}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$\left(-3.1,182^{\circ}\right)$
Identify and graph each polar equation. $r=3 \cos (4 \theta)$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees.The complex cube roots of $1+i$
Find a vector $\mathbf{v}$ whose magnitude is 4 and whose component in the $\mathbf{i}$ direction is twice the component in the $\mathbf{j}$ direction.
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$. $\mathbf{v}=3 \mathbf{i}-\mathbf{j}+2 \mathbf{k}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j}-\mathbf{k}$
Volume of a Parallelepiped A parallelepiped is a prism whose faces are all parallelograms. Let $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ be the vectors that define the parallelepiped shown in the figure. The volume $\mathrm{V}$ of the parallelepiped is given by the formula $V=|(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}|$.(GRAPH CANT COPY)Find the volume of a parallelepiped if the defining vectors are $\mathbf{A}=3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}, \mathbf{B}=2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$, and $\mathbf{C}=3 \mathbf{i}-6 \mathbf{j}-2 \mathbf{k}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point.$(6.3,3.8)$
Identify and graph each polar equation. $r^2=9 \cos (2 \theta)$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of $\sqrt{3}-i$
Find a vector $\mathbf{v}$ whose magnitude is 3 and whose component in the $\mathbf{i}$ direction is equal to the component in the $\mathbf{j}$ direction.
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$. $\mathbf{v}=\mathbf{i}+3 \mathbf{j}+2 \mathbf{k}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}+\mathbf{k}$
Volume of a Parallelepiped Refer to Problem 55. Find the volume of a parallelepiped whose defining vectors are $\mathbf{A}=\mathbf{i}+6 \mathbf{k}, \mathbf{B}=2 \mathbf{i}+3 \mathbf{j}-8 \mathbf{k}$, and $\mathbf{C}=8 \mathbf{i}-5 \mathbf{j}+6 \mathbf{k}$.
Polar coordinates of a point are given. Find the rectangular coordinates of each point. $(8.1,5.2)$
Identify and graph each polar equation.$r^2=\sin (2 \theta)$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of $4-4 \sqrt{3} i$
If $\mathbf{v}=2 \mathbf{i}-\mathbf{j}$ and $\mathbf{w}=x \mathbf{i}+3 \mathbf{j}$, find all numbers $x$ for which $\|\mathbf{v}+\mathbf{w}\|=5$.
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$.$\mathbf{v}=3 \mathbf{i}+4 \mathbf{j}+\mathbf{k}, \quad \mathbf{w}=6 \mathbf{i}+8 \mathbf{j}+2 \mathbf{k}$
Prove for vectors $\mathbf{u}$ and $\mathbf{v}$ that$$\|\mathbf{u} \times \mathbf{v}\|^2=\|\mathbf{u}\|^2\|\mathbf{v}\|^2-(\mathbf{u} \cdot \mathbf{v})^2$$
In Problems 57-68, the rectangular coordinates of a point are given. Find polar coordinates for each point.$(3,0)$
Identify and graph each polar equation.$r=2^\theta$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees.The complex cube roots of $-8-8 i$
If $P=(-3,1)$ and $Q=(x, 4)$, find all numbers $x$ such that the vector represented by $\overline{P Q}$ has length 5 .
In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$.$\mathbf{v}=3 \mathbf{i}-4 \mathbf{j}+\mathbf{k}, \quad \mathbf{w}=6 \mathbf{i}-8 \mathbf{j}+2 \mathbf{k}$
Show that if $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, then$$\|\mathbf{u} \times \mathbf{v}\|=\|\mathbf{u}\|\|\mathbf{v}\|$$
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(0,2)$
Identify and graph each polar equation. $r=3^\theta$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fourth roots of $-16 i$
In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis.$\|\mathbf{v}\|=5, \quad \alpha=60^{\circ}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7).$\mathbf{v}=3 \mathbf{i}-6 \mathbf{j}-2 \mathbf{k}$
Show that if $\mathbf{u}$ and $\mathbf{v}$ are orthogonal unit vectors, then $\mathbf{u} \times \mathbf{v}$ is also a unit vector.
The rectangular coordinates of a point are given. Find polar coordinates for each point. $(-1,0)$
Identify and graph each polar equation. $r=1-\cos \theta$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees.The complex cube roots of -8
In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis.$\|\mathbf{v}\|=8, \quad \alpha=45^{\circ}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $\mathbf{v}=-6 \mathbf{i}+12 \mathbf{j}+4 \mathbf{k}$
Prove property (3).
The rectangular coordinates of a point are given. Find polar coordinates for each point. $(0,-2)$
Identify and graph each polar equation.$r=3+\cos \theta$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees.The complex fifth roots of $i$
In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis. $\|\mathbf{v}\|=14, \quad \alpha=120^{\circ}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $\mathbf{v}=\mathbf{i}+\mathbf{j}+\mathbf{k}$
Prove property (5).
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(1,-1)$
Identify and graph each polar equation. $r=1-3 \cos \theta$
In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fifth roots of $-i$
In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis.$\|\mathbf{v}\|=3, \quad \alpha=240^{\circ}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $v=i-j-k$
Prove property (9).
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(-3,3)$
Identify and graph each polar equation. $r=4 \cos (3 \theta)$
Find the four complex fourth roots of unity (1) and plot them.
In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis. $\|\mathbf{v}\|=25, \quad \alpha=330^{\circ}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $v=i+j$
If $\mathbf{u} \cdot \mathbf{v}=0$ and $\mathbf{u} \times \mathbf{v}=\mathbf{0}$, what, if anything, can you conclude about $\mathbf{u}$ and $\mathbf{v}$ ?
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(\sqrt{3}, 1)$
In Problems 63-68, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.$r=8 \cos \theta ; r=2 \sec \theta$
Find the six complex sixth roots of unity (1) and plot them.
In Problems 59-64, write the vector $\mathbf{v}$ in the form a $+b \mathbf{j}$, given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$-axis.$\|\mathbf{v}\|=15, \quad \alpha=315^{\circ}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $v=j+k$
Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of $\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$.
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(-2,-2 \sqrt{3})$
Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $r=8 \sin \theta ; r=4 \csc \theta$
Show that each complex $n$th root of a nonzero complex number $w$ has the same magnitude.
In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=3 \mathbf{i}+3 \mathbf{j}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$
Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find two pairs of polar coordinates $(r, \theta)$, one with $r>0$ and the other with $r<0$, for the point with rectangular coordinates $(-8,-15)$. Express $\theta$ in radians.
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(1.3,-2.1)$
Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.$r=\sin \theta ; r=1+\cos \theta$
Use the result of Problem 65 to draw the conclusion that each complex $n$th root lies on a circle with center at the origin. What is the radius of this circle?
In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}$
In Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). $\mathbf{v}=2 \mathbf{i}+3 \mathbf{j}-4 \mathbf{k}$
Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. For $f(x)=7^{x-1}+5$, find $f^{-1}(x)$.
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(-0.8,-2.1)$
Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $r=3 ; r=2+2 \cos \theta$
Refer to Problem 66. Show that the complex $n$th roots of a nonzero complex number $w$ are equally spaced on the circle.
In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=-3 \sqrt{3} \mathbf{i}+3 \mathbf{j}$
Robotic Arm Consider the double-jointed robotic arm shown in the figure. Let the lower arm be modeled by $\mathbf{a}=\langle 2,3,4\rangle$, the middle arm be modeled by $\mathbf{b}=\langle 1,-1,3\rangle$, and the upper arm be modeled by $\mathbf{c}=\langle 4,-1,-2\rangle$, where units are in feet.(a) Find a vector $\mathbf{d}$ that represents the position of the hand.(b) Determine the distance of the hand from the origin.(IMAGE CANT COPY)
Problems 64-67 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Use properties of logarithms to write $\log _4 \frac{\sqrt{x}}{z^3}$ as a sum or difference of logarithms. Express powers as factors.
The rectangular coordinates of a point are given. Find polar coordinates for each point.$(8.3,4.2)$
Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.$r=1+\sin \theta ; r=1+\cos \theta$
Prove formula (6).
In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=-5 \mathbf{i}-5 \mathbf{j}$
The Sphere In space, the collection of all points that are the same distance from some fixed point is called a sphere. See the illustration. The constant distance is called the radius, and the fixed point is the center of the sphere. Show that an equation of a sphere with center at $\left(x_0, y_0, z_0\right)$ and radius $r$ is$$\left(x-x_0\right)^2+\left(y-y_0\right)^2+\left(z-z_0\right)^2=r^2$$(IMAGE CANT COPY)
The rectangular coordinates of a point are given. Find polar coordinates for each point. $(-2.3,0.2)$
Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $r=1+\cos \theta ; r=3 \cos \theta$
Prove that De Moivre's Theorem is true for all integers $n$ by assuming it is true for integers $n \geq 1$ and then showing it is true for 0 and for negative integers.
In Problems 65-72, find the direction angle of $\mathbf{v}$. $v=4 i-2 j$
In Problems 69 and 70 , find an equation of a sphere with radius $r$ and center $P_0$. $r=1 ; P_0=(3,1,1)$
In Problems 69-76, the letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$2 x^2+2 y^2=3$
In Problems 69-72, the polar equation for each graph is either $r=a+b \cos \theta$ or $r=a+b \sin \theta, a>0$. Select the correct equation and find the values of $a$ and $b$.(Figure Can't Copy)
Mandelbrot Sets(a) Consider the expression $a_n=\left(a_{n-1}\right)^2+z$, where $z$ is some complex number (called the seed) and $a_0=z$. Compute $a_1\left(=a_0^2+z\right), a_2\left(=a_1^2+z\right), a_3\left(=a_2^2+z\right), a_4, a_5$, and $a_6$ for the following seeds: $z_1=0.1-0.4 i$, $z_2=0.5+0.8 i, \quad z_3=-0.9+0.7 i, \quad z_4=-1.1+0.1 i$, $z_5=0-1.3 i$, and $z_6=1+1 i$.(b) The dark portion of the graph represents the set of all values $z=x+y i$ that are in the Mandelbrot set.Determine which complex numbers in part (a) are in this set by plotting them on the graph. Do the complex numbers that are not in the Mandelbrot set have any common characteristics regarding the values of $a_6$ found in part (a)?(c) Compute $|z|=\sqrt{x^2+y^2}$ for each of the complex numbers in part (a). Now compute $\left|a_6\right|$ for each of the complex numbers in part (a). For which complex numbers is $\left|a_6\right| \leq|z|$ and $|z| \leq 2$ ? Conclude that the criterion for a complex number to be in the Mandelbrot set is that $\left|a_n\right| \leq|z|$ and $|z| \leq 2$.(GRAPH CANT COPY)
In Problems 65-72, find the direction angle of $\mathbf{v}$.$\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}$
In Problems 69 and 70 , find an equation of a sphere with radius $r$ and center $P_0$.$r=2 ; P_0=(1,2,2)$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$x^2+y^2=x$
Problems 71-74 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of the triangle with $a=8, b=11$, and $C=113^{\circ}$.
In Problems 65-72, find the direction angle of $\mathbf{v}$.$\mathbf{v}=-\mathbf{i}-5 \mathbf{j}$
In Problems 71-76, find the radius and center of each sphere. $x^2+y^2+z^2+2 x-2 y=2$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$x^2=4 y$
Problems 71-74 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Convert $240^{\circ}$ to radians. Express your answer as a multiple of $\pi$.
In Problems 65-72, find the direction angle of $\mathbf{v}$. $\mathbf{v}=-\mathbf{i}+3 \mathbf{j}$
In Problems 71-76, find the radius and center of each sphere.$x^2+y^2+z^2+2 x-2 z=-1$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$. $y^2=2 x$
Problems 71-74 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact distance between the points $(-3,4)$ and $(2,-1)$.
Force Vectors A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of $30^{\circ}$ with the ground. Express the force vector $\mathbf{F}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.
In Problems 71-76, find the radius and center of each sphere. $x^2+y^2+z^2-4 x+4 y+2 z=0$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$2 x y=1$
In Problems 73-82, graph each polar equation.$r=\frac{2}{1-\cos \theta} \quad$ (parabola)
Problems 71-74 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Determine whether $f(x)=5 x^2-12 x+4$ has a maximum value or a minimum value, and then find the value.
Force Vectors A man pushes a wheelbarrow up an incline of $20^{\circ}$ with a force of 100 pounds. Express the force vector $\mathbf{F}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.
In Problems 71-76, find the radius and center of each sphere.$x^2+y^2+z^2-4 x=0$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$4 x^2 y=1$
Graph each polar equation. $r=\frac{2}{1-2 \cos \theta} \quad$ (hyperbola)
Resultant Force Two forces of magnitude 40 newtons (N) and $60 \mathrm{~N}$ act on an object at angles of $30^{\circ}$ and $-45^{\circ}$ with the positive $x$-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find $\mathbf{F}_1+\mathbf{F}_2$.(GRAPH CANT COPY)
In Problems 71-76, find the radius and center of each sphere. $2 x^2+2 y^2+2 z^2-8 x+4 z=-1$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$x=4$
Graph each polar equation.$r=\frac{1}{3-2 \cos \theta} \quad$ (ellipse)
Resultant Force Two forces of magnitude 30 newtons (N) and $70 \mathrm{~N}$ act on an object at angles of $45^{\circ}$ and $120^{\circ}$ with the positive $x$-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find $\mathbf{F}_1+\mathbf{F}_2$.(GRAPH CANT COPY)
In Problems 71-76, find the radius and center of each sphere. $3 x^2+3 y^2+3 z^2+6 x-6 y=3$
The letters $x$ and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$.$y=-3$
Graph each polar equation. $r=\frac{1}{1-\cos \theta} \quad$ (parabola)
Finding the Actual Speed and Direction of an Aircraft A Boeing 747 jumbo jet maintains a constant airspeed of 550 miles per hour $(\mathrm{mph})$ headed due north. The jet stream is $100 \mathrm{mph}$ in the northeasterly direction.(a) Express the velocity $\mathbf{v}_{\mathrm{a}}$ of the 747 relative to the air and the velocity $\mathbf{v}_{\mathrm{w}}$ of the jet stream in terms of $\mathbf{i}$ and $\mathbf{j}$.(b) Find the velocity of the 747 relative to the ground.(c) Find the actual speed and direction of the 747 relative to the ground.
The work $W$ done by a constant force $\mathbf{F}$ in moving an object from a point $A$ in space to a point $B$ in space is defined as $W=\mathbf{F} \cdot \overrightarrow{A B}$. Use this definition in Problems 77-79. Work Find the work done by a force of 3 newtons acting in the direction $2 \mathbf{i}+\mathbf{j}+2 \mathbf{k}$ in moving an object 2 meters from $(0,0,0)$ to $(0,2,0)$.
In Problems $77-84$, the letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$. $r=\cos \theta$
Graph each polar equation. $r=\theta, \quad \theta \geq 0 \quad$ (spiral of Archimedes)
Finding the Actual Speed and Direction of an Aircraft An Airbus A320 jet maintains a constant airspeed of $500 \mathrm{mph}$ headed due west. The jet stream is $100 \mathrm{mph}$ in the southeasterly direction.(a) Express the velocity $\mathbf{v}_{\mathrm{a}}$ of the A320 relative to the air and the velocity $\mathbf{v}_{\mathrm{w}}$ of the jet stream in terms of $\mathbf{i}$ and $\mathbf{j}$.(b) Find the velocity of the A320 relative to the ground.(c) Find the actual speed and direction of the A 320 relative to the ground.
The work $W$ done by a constant force $\mathbf{F}$ in moving an object from a point $A$ in space to a point $B$ in space is defined as $W=\mathbf{F} \cdot \overrightarrow{A B}$. Use this definition in Problems 77-79.Work Find the work done by a force of 1 newton acting in the direction $2 \mathbf{i}+2 \mathbf{j}+\mathbf{k}$ in moving an object 3 meters from $(0,0,0)$ to $(1,2,2)$.
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.$r=\sin \theta+1$
Graph each polar equation. $r=\frac{3}{\theta} \quad$ (reciprocal spiral)
Ground Speed and Direction of an Airplane An airplane has an airspeed of 500 kilometers per hour $(\mathrm{km} / \mathrm{h})$ bearing $\mathrm{N} 45^{\circ} \mathrm{E}$. The wind velocity is $60 \mathrm{~km} / \mathrm{h}$ in the direction $\mathrm{N} 30^{\circ} \mathrm{W}$. Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?
The work $W$ done by a constant force $\mathbf{F}$ in moving an object from a point $A$ in space to a point $B$ in space is defined as $W=\mathbf{F} \cdot \overrightarrow{A B}$. Use this definition in Problems 77-79.Work Find the work done in moving an object along a vector $\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}-5 \mathbf{k}$ if the applied force is $\mathbf{F}=2 \mathbf{i}-\mathbf{j}-\mathbf{k}$. Use meters for distance and newtons for force.
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.$r^2=\cos \theta$
Graph each polar equation.$r=\csc \theta-2, \quad 0<\theta<\pi \quad$ (conchoid)
Ground Speed and Direction of an Airplane An airplane has an airspeed of $600 \mathrm{~km} / \mathrm{h}$ bearing $\mathrm{S} 30^{\circ} \mathrm{E}$. The wind velocity is $40 \mathrm{~km} / \mathrm{h}$ in the direction $\mathrm{S} 45^{\circ} \mathrm{E}$. Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?
Problems $80-83$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: $\frac{3}{x-2} \geq 5$
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.$r=\sin \theta-\cos \theta$
Graph each polar equation. $r=\sin \theta \tan \theta \quad$ (cissoid)
Weight of a Boat A magnitude of 700 pounds of force is required to hold a boat and its trailer in place on a ramp whose incline is $10^{\circ}$ to the horizontal. What is the combined weight of the boat and its trailer?
Problems $80-83$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Given $f(x)=2 x-3$ and $g(x)=x^2+x-1$, find $(f \circ g)(x)$.
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$. $r=2$
Graph each polar equation.$r=\tan \theta,-\frac{\pi}{2}<\theta<\frac{\pi}{2} \quad$ (kappa curve)
Weight of a Car A magnitude of 1200 pounds of force is required to prevent a car from rolling down a hill whose incline is $15^{\circ}$ to the horizontal. What is the weight of the car?
Problems $80-83$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of $\sin 80^{\circ} \cos 50^{\circ}-\cos 80^{\circ} \sin 50^{\circ}$.
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$. $r=4$
Graph each polar equation. $r=\cos \frac{\theta}{2}$
Correct Direction for Crossing a River A river has a constant current of $3 \mathrm{~km} / \mathrm{h}$. At what angle to a boat dock should a motorboat capable of maintaining a constant speed of $20 \mathrm{~km} / \mathrm{h}$ be headed in order to reach a point directly opposite the dock? If the river is $\frac{1}{2}$ kilometer wide, how long will it take to cross?(IMAGE CANT COPY)
Problems $80-83$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve the triangle.(GRAPH CANT COPY)
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.$r=\frac{4}{1-\cos \theta}$
Show that the graph of the equation $r \sin \theta=a$ is a horizontal line $a$ units above the pole if $a \geq 0$ and $|a|$ units below the pole if $a<0$.
Finding the Correct Compass Heading The pilot of an aircraft wishes to head directly east but is faced with a wind speed of $40 \mathrm{mph}$ from the northwest. If the pilot maintains an airspeed of $250 \mathrm{mph}$, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft?
The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.$r=\frac{3}{3-\cos \theta}$
Show that the graph of the equation $r \cos \theta=a$ is a vertical line $a$ units to the right of the pole if $a \geq 0$ and $|a|$ units to the left of the pole if $a<0$.
Charting a Course A helicopter pilot needs to travel to a regional airport 25 miles away. She flies at an actual heading of $\mathrm{N} 16.26^{\circ} \mathrm{E}$ with an airspeed of $120 \mathrm{mph}$, and there is a wind blowing directly east at $20 \mathrm{mph}$.(a) Determine the compass heading that the pilot needs to reach her destination.(b) How long will it take her to reach her destination? Round to the nearest minute.
Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive $x$-axis.(a) Write the location of Wrigley Field using rectangular coordinates.(b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.(c) U.S. Cellular Field, home of the White Sox, is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates.(d) Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.
Show that the graph of the equation $r=2 a \sin \theta, a>0$, is a circle of radius $a$ with center at $(0, a)$ in rectangular coordinates.
Crossing a River A captain needs to pilot a boat across a river that is $2 \mathrm{~km}$ wide. The current in the river is $2 \mathrm{~km} / \mathrm{h}$ and the speed of the boat in still water is $10 \mathrm{~km} / \mathrm{h}$. The desired landing point on the other side is $1 \mathrm{~km}$ upstream.(a) Determine the direction in which the captain should aim the boat.(b) How long will the trip take?
Show that the formula for the distance $d$ between two points $P_1=\left(r_1, \theta_1\right)$ and $P_2=\left(r_2, \theta_2\right)$ is$$d=\sqrt{r_1^2+r_2^2-2 r_1 r_2 \cos \left(\theta_2-\theta_1\right)}$$
Show that the graph of the equation $r=-2 a \sin \theta, a>0$, is a circle of radius $a$ with center at $(0,-a)$ in rectangular coordinates.
Static Equilibrium A weight of 1000 pounds is suspended from two cables, as shown in the figure. What are the tensions in the two cables?(IMAGE CANT COPY)
In converting from polar coordinates to rectangular coordinates, what formulas will you use?
Show that the graph of the equation $r=2 a \cos \theta, a>0$, is a circle of radius $a$ with center at $(a, 0)$ in rectangular coordinates.
Static Equilibrium A weight of 800 pounds is suspended from two cables, as shown in the figure. What are the tensions in the two cables?(IMAGE CANT COPY)
Explain how to convert from rectangular coordinates to polar coordinates.
Show that the graph of the equation $r=-2 a \cos \theta, a>0$, is a circle of radius $a$ with center at $(-a, 0)$ in rectangular coordinates.
Static Equilibrium A tightrope walker located at a certain point deflects the rope as indicated in the figure. If the weight of the tightrope walker is 150 pounds, how much tension is in each part of the rope?(IMAGE CANT COPY)
Is the street system in your town based on a rectangular coordinate system, a polar coordinate system, or some other system? Explain.
Explain why the following test for symmetry is valid: Replace $r$ by $-r$ and $\theta$ by $-\theta$ in a polar equation. If an equivalent equation results, the graph is symmetric with respect to the line $\theta=\frac{\pi}{2}$ ( $y$-axis).(a) Show that the test on page 621 fails for $r^2=\cos \theta$, yet this new test works.(b) Show that the test on page 621 works for $r^2=\sin \theta$, yet this new test fails.
Static Equilibrium Repeat Problem 89 if the angle on the left is $3.8^{\circ}$, the angle on the right is $2.6^{\circ}$, and the weight of the tightrope walker is 135 pounds.
Problems $90-93$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Solve: $\log _4(x+3)-\log _4(x-1)=2$
Write down two different tests for symmetry with respect to the polar axis. Find examples in which one test works and the other fails. Which test do you prefer to use? Justify your answer.
Static Friction A 20-pound box sits at rest on a horizontal surface, and there is friction between the box and the surface. One side of the surface is raised slowly to create a ramp. The friction force $\mathbf{f}$ opposes the direction of motion and is proportional to the normal force $\mathbf{F}_{\mathrm{N}}$ exerted by the surface on the box. The proportionality constant is called the coefficient of friction, $\mu$. When the angle of the ramp, $\theta$, reaches $20^{\circ}$, the box begins to slide. Find the value of $\mu$ to two decimal places.(GRAPH CANT COPY)
Problems $90-93$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Use Descartes' Rule of Signs to determine the possible number of positive or negative real zeros for the function $f(x)=-2 x^3+6 x^2-7 x-8$.
The tests for symmetry given on page 621 are sufficient, but not necessary. Explain what this means.
Inclined Ramp A 2-pound weight is attached to a 3-pound weight by a rope that passes over an ideal pulley. The smaller weight hangs vertically, while the larger weight sits on a frictionless inclined ramp with angle $\theta$. The rope exerts a tension force $\mathbf{T}$ on both weights along the direction of the rope. Find the angle measure for $\theta$ that is needed to keep the larger weight from sliding down the ramp. Round your answer to the nearest tenth of a degree.(GRAPH CANT COPY)
Problems $90-93$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the midpoint of the line segment connecting the points $(-3,7)$ and $\left(\frac{1}{2}, 2\right)$.
Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.
Inclined Ramp A box sitting on a horizontal surface is attached to a second box sitting on an inclined ramp by a rope that passes over an ideal pulley. The rope exerts a tension force $\mathbf{T}$ on both weights along the direction of the rope, and the coefficient of friction between the surface and boxes is 0.6 (see Problems 91 and 92). If the box on the right weighs 100 pounds and the angle of the ramp is $35^{\circ}$, how much must the box on the left weigh for the system to be in static equilibrium? Round your answer to two decimal places.(GRAPH CANT COPY)
Problems $90-93$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Given that the point $(3,8)$ is on the graph of $y-f(x)$, what is the corresponding point on the graph of $y=-2 f(x+3)+5$ ?
Problems 93-96 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Solve: $\frac{5}{x-3} \geq 1$
Muscle Force Two muscles exert force on a bone at the same point. The first muscle exerts a force of $800 \mathrm{~N}$ at a $10^{\circ}$ angle with the bone. The second muscle exerts a force of $710 \mathrm{~N}$ at a $35^{\circ}$ angle with the bone. What are the direction and magnitude of the resulting force on the bone?
Problems 93-96 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Convert $\frac{7 \pi}{3}$ radians to degrees.
Truck Pull At a county fair truck pull, two pickup trucks are attached to the back end of a monster truck as illustrated in the figure. One of the pickups pulls with a force of 2000 pounds, and the other pulls with a force of 3000 pounds. There is an angle of $45^{\circ}$ between them. With how much force must the monster truck pull in order to remain unmoved?(IMAGE CANT COPY)
Problems 93-96 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine the amplitude and period of $y=-2 \sin (5 x)$
Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbor's tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a $40^{\circ}$ angle between the forces, as shown in the figure.(a) Assuming the farmer's estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump?(b) Had the farmer arranged the tractors with a $25^{\circ}$ angle between the forces, would he have been successful in removing the stump?(IMAGE CANT COPY)
Problems 93-96 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Find any asymptotes for the graph of without graphing.$$R(x)=\frac{x+3}{x^2-x-12} \text {. }$$
Computer Graphics The field of computer graphics utilizes vectors to compute translations of points. For example, if the point $(-3,2)$ is to be translated by $\mathbf{v}=\langle 5,2\rangle$, then the new location will be $\mathbf{u}^{\prime}=\mathbf{u}+\mathbf{v}=\langle-3,2\rangle+\langle 5,2\rangle=\langle 2,4\rangle$.As illustrated in the figure, the point $(-3,2)$ is translated to $(2,4)$ by $\mathbf{v}$.(a) Determine the new coordinates of $(3,-1)$ if it is translated by $\mathbf{v}=\langle-4,5\rangle$.(b) Illustrate this translation graphically.(GRAPH CANT COPY)
Computer Graphics Refer to Problem 97. The points $(-3,0),(-1,-2),(3,1)$, and $(1,3)$ are the vertices of a parallelogram $A B C D$.(a) Find the new vertices of a parallelogram $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ if it is translated by $\mathbf{v}=\langle 3,-2\rangle$.(b) Find the new vertices of a parallelogram $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ if it is translated by $-\frac{1}{2} \mathbf{v}$.
Static Equilibrium Show on the following graph the force needed for the object at $P$ to be in static equilibrium.(GRAPH CANT COPY)
Explain in your own words what a vector is. Give an example of a vector.
Write a brief paragraph comparing the algebra of complex numbers and the algebra of vectors.
Explain the difference between an algebraic vector and a position vector.
Problems 103-106 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Solve triangle $A B C: a=4, b=1$, and $C=100^{\circ}$
Problems 103-106 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the real zeros of $f(x)=-3 x^3+12 x^2+36 x$.
Problems 103-106 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of $\tan \left[\cos ^{-1}\left(\frac{1}{2}\right)\right]$.
Problems 103-106 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Find the amplitude, period, and phase shift of $y=\frac{3}{2} \cos (6 x+3 \pi)$. Graph the function, showing at least two periods.