Calculus Volume 3

Gilbert Strang

Chapter 2

Vectors in Space - all with Video Answers

Educators

Section 1

Vectors in the Plane

Problem 1

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
\overrightarrow{P Q}
$$

Jie Min

Jie Min

Numerade Educator

Problem 2

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
\overrightarrow{P R}
$$

Amy Jiang

Numerade Educator

Problem 3

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
\overrightarrow{Q P}
$$

Jie Min

Numerade Educator

Problem 4

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
\overrightarrow{R P}
$$

Amy Jiang

Numerade Educator

Problem 5

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
\overrightarrow{P Q}+\overrightarrow{P R}
$$

Jie Min

Numerade Educator

Problem 6

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
\overrightarrow{P Q}-\overrightarrow{P R}
$$

Amy Jiang

Numerade Educator

Problem 7

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
2 \overrightarrow{P Q}-2 \overrightarrow{P R}
$$

Jie Min

Numerade Educator

Problem 8

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
$$
2 \overrightarrow{P Q}+\frac{1}{2} \overrightarrow{P R}
$$

Amy Jiang

Numerade Educator

Problem 9

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
The unit vector in the direction of $\overrightarrow{P Q}$

Jie Min

Numerade Educator

Problem 10

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
The unit vector in the direction of $\overrightarrow{P R}$

Amy Jiang

Numerade Educator

Problem 11

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
A vector $\mathbf{v}$ has initial point $(-1,-3)$ and terminal point $(2,1) .$ Find the unit vector in the direction of $\mathbf{v} .$ Express the answer in component form.

Jie Min

Numerade Educator

Problem 12

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
A vector v has initial point $(-2,5)$ and terminal point $(3,-1)$ . Find the unit vector in the direction of $v$ Express the answer in component form.

Amy Jiang

Numerade Educator

Problem 13

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
The vector $\mathbf{v}$ has initial point $P(1,0)$ and terminal point $Q$ that is on the $y$ -axis and above the initial point. Find the coordinates of terminal point $Q$ such that the magnitude of the vector $\mathbf{v}$ is $\sqrt{5}$ .

Jie Min

Numerade Educator

Problem 14

For the following exercises, consider points $P(-1,3)$ $Q(1,5),$ and $R(-3,7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.
The vector $\mathbf{v}$ has initial point $P(1,1)$ and terminal point $Q$ that is on the $x$ -axis and left of the initial point. Find the coordinates of terminal point $Q$ such that the magnitude of the vector $\mathbf{v}$ is $\sqrt{10}$ .

Amy Jiang

Numerade Educator

Problem 15

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
$$
\mathbf{a}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{b}=\mathbf{i}+3 \mathbf{j}
$$

Jie Min

Numerade Educator

Problem 16

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
$$
\mathbf{a}=2 \mathbf{i}, \quad \mathbf{b}=-2 \mathbf{i}+2 \mathbf{j}
$$

Amy Jiang

Numerade Educator

Problem 17

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Let a be a standard-position vector with terminal point $(-2,-4)$ . Let $\mathbf{b}$ be a vector with initial point $(1,2)$ and terminal point $(-1,4) .$ Find the magnitude of vector $-3 \mathbf{a}+\mathbf{b}-4 \mathbf{i}+\mathbf{j}$

Jie Min

Numerade Educator

Problem 18

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Let a be a standard-position vector with terminal point at $(2,5) .$ Let $\mathbf{b}$ be a vector with initial point $(-1,3)$ and terminal point $(1,0) .$ Find the magnitude of vector a $-3 \mathbf{b}+14 \mathbf{i}-14 \mathbf{j}$ .

Amy Jiang

Numerade Educator

Problem 19

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Let $\mathbf{u}$ and $\mathbf{v}$ be two nonzero vectors that are nonequivalent. Consider the vectors $\mathbf{a}=4 \mathbf{u}+5 \mathbf{v}$ and $\mathbf{b}=\mathbf{u}+2 \mathbf{v}$ defined in terms of $\mathbf{u}$ and $\mathbf{v}$ . Find the scalar $\lambda$ such that vectors $\mathbf{a}+\lambda \mathbf{b}$ and $\mathbf{u}-\mathbf{v}$ are equivalent.

Jie Min

Numerade Educator

Problem 20

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Let $\mathbf{u}$ and $\mathbf{v}$ be two nonzero vectors that are nonequivalent. Consider the vectors $\mathbf{a}=2 \mathbf{u}-4 \mathbf{v}$ and $\mathbf{b}=3 \mathbf{u}-7 \mathbf{v}$ defined in terms of $\mathbf{u}$ and $\mathbf{v}$ . Find the scalars $\alpha$ and $\beta$ such that vectors $\alpha \mathbf{a}+\beta \mathbf{b}$ and $\mathbf{u}-\mathbf{v}$ are equivalent.

Amy Jiang

Numerade Educator

Problem 21

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Consider the vector $\quad \mathbf{a}(t)=\langle\cos t, \sin t\rangle \quad$ with components that depend on a real number $t$ . As the number $t$ varies, the components of $\mathbf{a}(t)$ change as well, depending on the functions that define them.
a. Write the vectors $\mathbf{a}(0)$ and $\mathbf{a}(\pi)$ in component form.
b. Show that the magnitude $\|\mathbf{a}(t)\|$ of vector $\mathbf{a}(t)$ remains constant for any real number $t$
c. As $t$ varies, show that the terminal point of vector $\mathbf{a}(t)$ describes a circle centered at the origin of radius $1 .$

Jie Min

Numerade Educator

Problem 22

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Consider vector $\quad \mathbf{a}(x)=\left\langle x, \sqrt{1-x^{2}}\right\rangle \quad$ with components that depend on a real number $x \in[-1,1]$ . As the number $x$ varies, the components of $\mathbf{a}(x)$ change as well, depending on the functions that define them.
a. Write the vectors a(0) and a(1) in component form.
b. Show that the magnitude $\|\mathrm{a}(x)\|$ of vector a(x) remains constant for any real number $x$
c. As $x$ varies, show that the terminal point of vector a(x) describes a circle centered at the origin of
radius 1 .

Amy Jiang

Numerade Educator

Problem 23

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Show that vectors $\quad \mathbf{a}(t)=\langle\cos t, \sin t\rangle \quad$ and $\mathbf{a}(x)=\left\langle x, \sqrt{1-x^{2}}\right\rangle \quad$ are equivalent for $\quad x=r$ and $t=2 k \pi,$ where $k$ is an integer.

Jie Min

Numerade Educator

Problem 24

For the following exercises, use the given vectors a and b.
a. Determine the vector sum $\mathbf{a}+\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
b. Find the vector difference $\mathbf{a}-\mathbf{b}$ and express it in both the component form and by using the standard unit vectors.
c. Verify that the vectors a, $\mathbf{b},$ and $\mathbf{a}+\mathbf{b},$ and, respectively, a, $\mathbf{b},$ and $\mathbf{a}-\mathbf{b}$ satisfy the triangle inequality.
d. Determine the vectors $2 \mathrm{a}, \quad-\mathrm{b},$ and $2 \mathrm{a} \mathrm{b}$ . Express the vectors in both the component form and by using standard unit vectors.
Show that vectors $\quad \mathbf{a}(t)=\langle\cos t, \sin t\rangle \quad$ and $\mathbf{a}(x)=\left\langle x, \sqrt{1-x^{2}}\right\rangle \quad$ are opposite for $\quad x=r$ and $t=\pi+2 k \pi,$ where $k$ is an integer.

Amy Jiang

Numerade Educator

Problem 25

For the following exercises, find vector $\mathbf{v}$ with the given magnitude and in the same direction as vector $\mathbf{u} .$
$$
\|\mathbf{v}\|=7, \mathbf{u}=\langle 3,4\rangle
$$

Jie Min

Numerade Educator

Problem 26

For the following exercises, find vector $\mathbf{v}$ with the given magnitude and in the same direction as vector $\mathbf{u} .$
$$
\|\mathbf{v}\|=3, \mathbf{u}=\langle- 2,5\rangle
$$

Amy Jiang

Numerade Educator

Problem 27

For the following exercises, find vector $\mathbf{v}$ with the given magnitude and in the same direction as vector $\mathbf{u} .$
$$
\|\mathbf{v}\|=7, \mathbf{u}=\langle 3,-5\rangle
$$

Jie Min

Numerade Educator

Problem 28

For the following exercises, find vector $\mathbf{v}$ with the given magnitude and in the same direction as vector $\mathbf{u} .$
$$
\|\mathbf{v}\|=10, \mathbf{u}=\langle 2,-1\rangle
$$

Amy Jiang

Numerade Educator

Problem 29

For the following exercises, find the component form of vector $\mathbf{u},$ given its magnitude and the angle the vector makes with the positive $x$ -axis. Give exact answers when possible.
$$
\|\mathbf{u}\|=2, \quad \theta=30^{\circ}
$$

Jie Min

Numerade Educator

Problem 30

For the following exercises, find the component form of vector $\mathbf{u},$ given its magnitude and the angle the vector makes with the positive $x$ -axis. Give exact answers when possible.
$$
\|\mathbf{u}\|=6, \quad \theta=60^{\circ}
$$

Amy Jiang

Numerade Educator

Problem 31

For the following exercises, find the component form of vector $\mathbf{u},$ given its magnitude and the angle the vector makes with the positive $x$ -axis. Give exact answers when possible.
$$
\|\mathbf{u}\|=5, \quad \theta=\frac{\pi}{2}
$$

Jie Min

Numerade Educator

Problem 32

For the following exercises, find the component form of vector $\mathbf{u},$ given its magnitude and the angle the vector makes with the positive $x$ -axis. Give exact answers when possible.
$$
\|\mathbf{u}\|=8, \quad \theta=\pi
$$

Amy Jiang

Numerade Educator

Problem 33

For the following exercises, find the component form of vector $\mathbf{u},$ given its magnitude and the angle the vector makes with the positive $x$ -axis. Give exact answers when possible.
$$
\|\mathbf{u}\|=10, \quad \theta=\frac{5 \pi}{6}
$$

Jie Min

Numerade Educator

Problem 34

For the following exercises, find the component form of vector $\mathbf{u},$ given its magnitude and the angle the vector makes with the positive $x$ -axis. Give exact answers when possible.
$$
\|\mathbf{u}\|=50, \quad \theta=\frac{3 \pi}{4}
$$

Amy Jiang

Numerade Educator

Problem 35

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
$$
\mathbf{u}=5 \sqrt{2} \mathbf{i}-5 \sqrt{2} \mathbf{j}
$$

Jie Min

Numerade Educator

Problem 36

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
$$
\mathbf{u}=-\sqrt{3} \mathbf{i}-\mathbf{j}
$$

Khushbu Rani

Numerade Educator

Problem 37

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Let $\quad \mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle, \quad \mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle, \quad$ and $\mathbf{c}=\left\langle c_{1}, c_{2}\right\rangle \quad$ be three nonzero vectors. If $a_{1} b_{2}-a_{2} b_{1} \neq 0,$ then show there two scalars, $\alpha$ and $\beta,$ such that $\mathbf{c}=\alpha \mathbf{a}+\beta \mathbf{b}$

Jie Min

Numerade Educator

Problem 38

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Consider vectors $\mathbf{a}=\langle 2,-4\rangle, \quad \mathbf{b}=\langle- 1,2\rangle$ and $\mathbf{0}$ Determine the scalars $\alpha$ and $\beta$ such that $\mathbf{c}=\alpha \mathbf{a}+\beta \mathbf{b}$

Amy Jiang

Numerade Educator

Problem 39

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Let $P\left(x_{0}, f\left(x_{0}\right)\right)$ be a fixed point on the graph of the differential function $f$ with a domain that is the set of real numbers.
a. Determine the real number $z_{0}$ such that point $Q\left(x_{0}+1, z_{0}\right)$ is situated on the line tangent to the graph of $f$ at point $P$ .
b. Determine the unit vector u with initial point $P$ and terminal point $Q .$

Jie Min

Numerade Educator

Problem 40

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Consider the function $f(x)=x^{4},$ where $x \in \mathbb{R}$
a. Determine the real number $z_{0}$ such that point $Q\left(2, z_{0}\right)$ s situated on the line tangent to the graph of $f$ at point $P(1,1)$ .
b. Determine the unit vector u with initial point $P$ and terminal point $Q$ .

Amy Jiang

Numerade Educator

Problem 41

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Consider $f$ and $g$ two functions defined on the same set of real numbers $D .$ Let $\quad \mathbf{a}=\langle x, f(x)\rangle$ and $\mathbf{b}=\langle x, g(x)\rangle$ be two vectors that describe the graphs of the functions, where $x \in D .$ Show that if the graphs of the functions $f$ and $g$ do not intersect, then the vectors a and $\mathbf{b}$ are not equivalent.

Jie Min

Numerade Educator

Problem 42

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Find $x \in \mathbb{R}$ such that vectors $\mathbf{a}=\langle x, \sin x\rangle$ and $\mathbf{b}=\langle x, \cos x\rangle$ are equivalent.

Amy Jiang

Numerade Educator

Problem 43

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Calculate the coordinates of point $D$ such that $A B C D$ is a parallelogram, with $A(1,1), \quad B(2,4),$ and $C(7,4)$

Jie Min

Numerade Educator

Problem 44

For the following exercises, vector $\mathbf{u}$ is given. Find the angle $\theta \in[0,2 \pi)$ that vector $\mathbf{u}$ makes with the positive direction of the $x$ -axis, in a counter-clockwise direction.
Consider the points $A(2,1), \quad B(10,6), \quad C(13,4)$ and $D(16,-2) .$ Determine the component form of vector $\overrightarrow{A D}$

Amy Jiang

Numerade Educator

Problem 45

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of 70 $\mathrm{mph}$ and an angle of elevation of $30^{\circ} .$ Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

Jie Min

Numerade Educator

Problem 46

A baseball player throws a baseball at an angle of $30^{\circ}$ with the horizontal. If the initial speed of the ball is 100 mph, find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)

Amy Jiang

Numerade Educator

Problem 47

A bullet is fired with an initial velocity of 1500 $\mathrm{ft} / \mathrm{sec}$ at an angle of $60^{\circ}$ with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet.
(Round to two decimal places.)

Khushbu Rani

Numerade Educator

Problem 48

A 65 -kg sprinter exerts a force of 798 N at a $19^{\circ}$ angle with respect to the ground on the starting block at the instant a race begins. Find the horizontal component of the force. (Round to two decimal places.)

Amy Jiang

Numerade Educator

Problem 49

Two forces, a horizontal force of 45 $\mathrm{lb}$ and another of 52 lb, act on the same object. The angle between these forces is $25^{\circ} .$ Find the magnitude and direction angle from the positive $x$ -axis of the resultant force that acts on the object. (Round to two decimal places.)

Jie Min

Numerade Educator

Problem 50

Two forces, a vertical force of 26 lb and another of 45 lb, act on the same object. The angle between these forces is $55^{\circ} .$ Find the magnitude and direction angle from the positive $x$ -axis of the resultant force that acts on the object. (Round to two decimal places.)

Khushbu Rani

Numerade Educator

Problem 51

Three forces act on object. Two of the forces have the magnitudes 58 $\mathrm{N}$ and $27 \mathrm{N},$ and make angles $53^{\circ}$ and $152^{\circ},$ respectively, with the positive $x$ -axis. Find the magnitude and the direction angle from the positive $x$ -axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

Jie Min

Numerade Educator

Problem 52

Three forces with magnitudes 80 lb, 120 lb, and 60 lb act on an object at angles of $45^{\circ}, \quad 60^{\circ}$ and $30^{\circ},$ respectively, with the positive $x$ -axis. Find the magnitude and direction angle from the positive $x$ -axis of the resultant force. (Round to two decimal places.)

Amy Jiang

Numerade Educator

Problem 53

An airplane is flying in the direction of $43^{\circ}$ east of north (also abbreviated as $\mathrm{N} 43 \mathrm{E} )$ at a speed of 550 $\mathrm{mph}$ . A wind with speed 25 $\mathrm{mph}$ comes from the southwest at a bearing of $\mathrm{N} 15 \mathrm{E}$ . What are the ground speed and new
direction of the airplane?

Jie Min

Numerade Educator

Problem 54

A boat is traveling in the water at 30 $\mathrm{mph}$ in a direction of $\mathrm{N} 20 \mathrm{E}$ (that is, $20^{\circ}$ east of north). A strong current is moving at 15 $\mathrm{mph}$ in a direction of $\mathrm{N} 45 \mathrm{E}$ . What are the new speed and direction of the boat?

Amy Jiang

Numerade Educator

Problem 55

A $50-$ lb weight is hung by a cable so that the two portions of the cable make angles of $40^{\circ}$ and $53^{\circ},$ respectively, with the horizontal. Find the magnitudes of the forces of tension $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

Khushbu Rani

Numerade Educator

Problem 56

A 62 -lb weight hangs from a rope that makes the angles of $29^{\circ}$ and $61^{\circ},$ respectively, with the horizontal. Find the magnitudes of the forces of tension $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

Amy Jiang

Numerade Educator

Problem 57

A $1500-1 b$ boat is parked on a ramp that makes an angle of $30^{\circ}$ with the horizontal. The boat's weight vector points downward and is a sum of two vectors: a horizontal vector $\mathbf{v}_{1}$ that is parallel to the ramp and a vertical vector $\mathbf{v}_{2}$ that is perpendicular to the inclined surface. The magnitudes of vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are the horizontal and vertical component, respectively, of the boat's weight vector. Find the magnitudes of $\mathbf{v}_{1}$ and $\mathbf{v}_{2} .$ (Round to the nearest integer.)

Jie Min

Numerade Educator

Problem 58

An $85-$ lb box is at rest on a $26^{\circ}$ incline. Determine the magnitude of the force parallel to the incline necessary to keep the box from sliding. (Round to the nearest integer.)

Amy Jiang

Numerade Educator

Problem 59

A guy-wire supports a pole that is 75 ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground 50 ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is 50 lb. (Round to the nearest integer.)

Jie Min

Numerade Educator

Problem 60

A telephone pole guy-wire has an angle of elevation of $35^{\circ}$ with respect to the ground. The force of tension in the guy-wire is 120 lb. Find the horizontal and vertical components of the force of tension. (Round to the nearest integer.)

Amy Jiang

Numerade Educator