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CLP-3 Multivariable Calculus 3

Joel Feldman, Andrew Rechnitzer and Elyse Yeager

Chapter 1

Vectors and Geometry in Two and Three Dimensions - all with Video Answers

Educators

Chapter Questions

02:19

Problem 1

Describe the set of all points $(x, y, z)$ in $\mathbb{R}^3$ that satisfy
1. $x^2+y^2+z^2=2 x-4 y+4$
2. $x^2+y^2+z^2<2 x-4 y+4$

Linh Vu
Linh Vu
Numerade Educator
01:26

Problem 1

Let $\mathbf{a}=\langle 2,0\rangle$ and $\mathbf{b}=\langle 1,1\rangle$. Evaluate and sketch $\mathbf{a}+\mathbf{b}, \mathbf{a}+2 \mathbf{b}$ and $2 \mathbf{a}-\mathbf{b}$.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:56

Problem 1

A line in $\mathbb{R}^2$ has direction $\mathbf{d}$ and passes through point $\mathbf{c}$.
Which of the following gives its parametric equation: $\langle x, y\rangle=\mathbf{c}+t \mathbf{d}$, or $\langle x, y\rangle=\mathbf{c}-t \mathbf{d}$ ?

Lucas Finney
Lucas Finney
Numerade Educator
02:02

Problem 1

The vector $\hat{\mathbf{k}}$ is a normal vector (i.e. is perpendicular) to the plane $z=0$. Find another nonzero vector that is normal to $z=0$.

Kamalesh Kumar
Kamalesh Kumar
Numerade Educator
02:03

Problem 1

What is wrong with the following exercise?
"Give an equation for the line passing through the point $(3,1,3)$ that is normal to the vectors $\langle 4,-6,2\rangle$ and $\left\langle\frac{1}{3},-\frac{1}{2}, \frac{1}{6}\right\rangle$."

Linh Vu
Linh Vu
Numerade Educator

Problem 1

Consider the following time-parametrized curve:

$$
\vec{r}(t)=\left(\cos \left(\frac{\pi}{4} t\right),(t-5)^2\right)
$$

List the three points $(-1 / \sqrt{2}, 0),(1,25)$, and $(0,25)$ in chronological order.

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01:10

Problem 1

Match the following equations and expressions with the corresponding pictures. Cartesian coordinates are ( $x, y, z$ ), cylindrical coordinates are ( $r, \theta, z$ ), and spherical coordinates are ( $\rho, \theta, \vec{a} r p h i$ ).
Figure a-f can't copy
(a) $\vec{a} r p h i=\pi / 3$
(b) $r=2 \cos \theta$
(c) $x^2+y^2=z^2+1$
(d) $\quad y=x^2+z^2$
(e) $\rho=2 \cos \vec{a} r p h i$
(f) $z=x^4+y^4-4 x y$

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 2

Describe and sketch the set of all points $(x, y)$ in $\mathbb{R}^2$ that satisfy
1. $x=y$
2. $x+y=1$
3. $x^2+y^2=4$
4. $x^2+y^2=2 y$
5. $x^2+y^2<2 y$

Carson Merrill
Carson Merrill
Numerade Educator
01:04

Problem 2

Determine whether or not the given points are collinear (that is, lie on a common straight line)
1. $(1,2,3),(0,3,7),(3,5,11)$
2. $(0,3,-5),(1,2,-2),(3,0,4)$

Carson Merrill
Carson Merrill
Numerade Educator
01:56

Problem 2

A line in $\mathbb{R}^2$ has direction $\mathbf{d}$ and passes through point $\mathbf{c}$.
Which of the following gives its parametric equation: $\langle x, y\rangle=\mathbf{c}+t \mathbf{d}$, or $\langle x, y\rangle=-\mathbf{c}+t \mathbf{d}$ ?

Lucas Finney
Lucas Finney
Numerade Educator
01:08

Problem 2

Consider the plane $P$ with equation $3 x+\frac{1}{2} y+z=4$.
1. Find the intersection of $P$ with the $y$-axis.
2. Find the intersection of $P$ with the $z$-axis.
3. Sketch the part of the intersection of $P$ with the $y z$-plane that is in the first octant. (That is, with $x, y, z \geq 0$.)

AG
Ankit Gupta
Numerade Educator
00:16

Problem 2

Find, if possible, four lines in 3d with
- no two of the lines parallel to each other and
- no two of the lines intersecting.

Gianna Calciano
Gianna Calciano
Numerade Educator

Problem 2

At what points in the $x y$-plane does the curve $\left(\sin t, t^2\right)$ cross itself? What is the difference in $t$ between the first time the curve crosses through a point, and the last?

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01:04

Problem 2

In each of (a) and (b) below, you are provided with a sketch of the first quadrant parts of a few level curves of some function $f(x, y)$. Sketch the first octant part of the corresponding graph $z=f(x, y)$.
Graph a-b can't copy

Carson Merrill
Carson Merrill
Numerade Educator

Problem 3

Describe the set of all points $(x, y, z)$ in $\mathbb{R}^3$ that satisfy the following conditions. Sketch the part of the set that is in the first octant.
1. $z=x$
2. $x+y+z=1$
3. $x^2+y^2+z^2=4$
4. $x^2+y^2+z^2=4, \quad z=1$
5. $x^2+y^2=4$
6. $z=x^2+y^2$

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01:01

Problem 3

Determine whether the given pair of vectors is perpendicular
1. $\langle 1,3,2\rangle,\langle 2,-2,2\rangle$
2. $\langle-3,1,7\rangle,\langle 2,-1,1\rangle$
3. $\langle 2,1,1\rangle,\langle-1,4,2\rangle$

Thane Stiles
Thane Stiles
Numerade Educator
01:23

Problem 3

Two points determine a line. Verify that the equations

$$
\langle x-1, y-9\rangle=t\langle 8,4\rangle
$$

and

$$
\langle x-9, y-13\rangle=t\left\langle 1, \frac{1}{2}\right\rangle
$$

describe the same line by finding two different points that lie on both lines.

Steven Clarke
Steven Clarke
Numerade Educator

Problem 3

1. Find the equation of the plane that passes through the origin and has normal vector $\langle 1,2,3\rangle$.
2. Find the equation of the plane that passes through the point $(0,0,1)$ and has normal vector $\langle 1,1,3\rangle$.
3. Find, if possible, the equation of a plane that passes through both $(1,2,3)$ and $(1,0,0)$ and has normal vector $\langle 4,5,6\rangle$.
4. Find, if possible, the equation of a plane that passes through both $(1,2,3)$ and $(0,3,4)$ and has normal vector $\langle 2,1,1\rangle$.

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01:43

Problem 3

Find a vector parametric equation for the line of intersection of the given planes.
1. $x-2 z=3$ and $y+\frac{1}{2} z=5$
2. $2 x-y-2 z=-3$ and $4 x-3 y-3 z=-5$

WM
William Mead
Numerade Educator

Problem 3

Find the specified parametrization of the first quadrant part of the circle $x^2+y^2=a^2$.
1. In terms of the $y$ coordinate.
2. In terms of the angle between the tangent line and the positive $x$-axis.
3. In terms of the arc length from $(0, a)$.

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01:06

Problem 3

Sketch a few level curves for the function $f(x, y)$ whose graph $z=f(x, y)$ is sketched below.
Graph can't copy

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:41

Problem 4

Let $A$ be the point $(2,1,3)$.
1. Find the distance from $A$ to the $x y$-plane.
2. Find the distance from $A$ to the $x z$-plane.
3. Find the distance from $A$ to the point $(x, 0,0)$ on the $x$-axis.
4. Find the point on the $x$-axis that is closest to $A$.
5. What is the distance from $A$ to the $x$-axis?

Carson Merrill
Carson Merrill
Numerade Educator
03:22

Problem 4

Consider the vector $\mathbf{a}=\langle 3,4\rangle$.
1. Find a unit vector in the same direction as $\mathbf{a}$.
2. Find all unit vectors that are parallel to $\mathbf{a}$.
3. Find all vectors that are parallel to $\mathbf{a}$ and have length 10 .
4. Find all unit vectors that are perpendicular to $\mathbf{a}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:51

Problem 4

A line in $\mathbb{R}^2$ has parametric equations

$$
\begin{aligned}
& x-3=9 t \\
& y-5=7 t
\end{aligned}
$$

There are many different ways to write the parametric equations of this line. If we rewrite the equations as

$$
\begin{aligned}
& x-x_0=d_x t \\
& y-y_0=d_y t
\end{aligned}
$$

what are all possible values of $\left\langle x_0, y_0\right\rangle$ and $\left\langle d_x, d_y\right\rangle$ ?

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:01

Problem 4

Find the equation of the plane that contains $(1,0,0),(0,1,0)$ and $(0,0,1)$.

Raj Bala
Raj Bala
Numerade Educator
01:19

Problem 4

Determine a vector equation for the line of intersection of the planes
1. $x+y+z=3$ and $x+2 y+3 z=7$
2. $x+y+z=3$ and $2 x+2 y+2 z=7$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:15

Problem 4

Diagram can't copy
A circle of radius $a$ rolls along the $x$-axis in the positive direction, starting with its centre at $(a, a)$. In that position, we mark the topmost point on the circle $P$. As the circle moves, $P$ moves with it. Let $\theta$ be the angle the circle has rolled - see the diagram below.
1. Give the position of the centre of the circle as a function of $\theta$.
2. Give the position of $P$ a function of $\theta$.
Diagram can't copy

Uma Kumari
Uma Kumari
Numerade Educator
03:44

Problem 4

Sketch some of the level curves of
1. $f(x, y)=x^2+2 y^2$
2. $f(x, y)=x y$
3. $f(x, y)=x e^{-y}$

Abhijith V
Abhijith V
Numerade Educator
03:47

Problem 5

Consider any triangle. Pick a coordinate system so that one vertex is at the origin and a second vertex is on the positive $x$-axis. Call the coordinates of the second vertex $(a, 0)$ and those of the third vertex $(b, c)$. Find the circumscribing circle (the circle that goes through all three vertices).

AG
Ankit Gupta
Numerade Educator
00:50

Problem 5

Consider the vector $\mathbf{b}=\langle 3,4,0\rangle$.
1. Find a unit vector in the same direction as $\mathbf{b}$.
2. Find all unit vectors that are parallel to $\mathbf{b}$.
3. Find four different unit vectors that are perpendicular to $\mathbf{b}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:14

Problem 5

Find the vector parametric, scalar parametric and symmetric equations for the line containing the given point and with the given direction.
1. point $(1,2)$, direction $\langle 3,2\rangle$
2. point $(5,4)$, direction $\langle 2,-1\rangle$
3. point $(-1,3)$, direction $\langle-1,2\rangle$

Linh Vu
Linh Vu
Numerade Educator
02:54

Problem 5

1. Find the equation of the plane containing the points $(1,0,1),(1,1,0)$ and $(0,1,1)$.
2. Is the point $(1,1,1)$ on the plane?
3. Is the origin on the plane?
4. Is the point $(4,-1,-1)$ on the plane?

Steven Clarke
Steven Clarke
Numerade Educator
04:42

Problem 5

In each case, determine whether or not the given pair of lines intersect. Also find all planes containing the pair of lines.
1. $\langle x, y, z\rangle=\langle-3,2,4\rangle+t\langle-4,2,1\rangle$ and $\langle x, y, z\rangle=\langle 2,1,2\rangle+t\langle 1,1,-1\rangle$
2. $\langle x, y, z\rangle=\langle-3,2,4\rangle+t\langle-4,2,1\rangle$ and $\langle x, y, z\rangle=\langle 2,1,-1\rangle+t\langle 1,1,-1\rangle$
3. $\langle x, y, z\rangle=\langle-3,2,4\rangle+t\langle-2,-2,2\rangle$ and $\langle x, y, z\rangle=\langle 2,1,-1\rangle+t\langle 1,1,-1\rangle$
4. $\langle x, y, z\rangle=\langle 3,2,-2\rangle+t\langle-2,-2,2\rangle$ and $\langle x, y, z\rangle=\langle 2,1,-1\rangle+t\langle 1,1,-1\rangle$

Linda Hand
Linda Hand
Numerade Educator
01:38

Problem 5

The curve $C$ is defined to be the intersection of the ellipsoid

$$
x^2-\frac{1}{4} y^2+3 z^2=1
$$

and the plane

$$
x+y+z=0
$$

When $y$ is very close to 0 , and $z$ is negative, find an expression giving $z$ in terms of $y$.

Gopesh Vishwakarma
Gopesh Vishwakarma
Numerade Educator

Problem 5

Sketch the level curves of $f(x, y)=\frac{2 y}{x^2+y^2}$.

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03:28

Problem 6

A certain surface consists of all points $P=(x, y, z)$ such that the distance from $P$ to the point $(0,0,1)$ is equal to the distance from $P$ to the plane $z+1=0$. Find an equation for the surface, sketch and describe it verbally.

Patrick Vaughn
Patrick Vaughn
Numerade Educator
04:16

Problem 6

Let $\mathbf{a}=\left\langle a_1, a_2\right\rangle$. Compute the projection of $\mathbf{a}$ on $\hat{\imath}$ and $\hat{\jmath}$.

Victor Salazar
Victor Salazar
Numerade Educator
00:45

Problem 6

Find the vector parametric, scalar parametric and symmetric equations for the line containing the given point and with the given normal.
1. point $(1,2)$, normal $\langle 3,2\rangle$
2. point $(5,4)$, normal $\langle 2,-1\rangle$
3. point $(-1,3)$, normal $\langle-1,2\rangle$

Jie Min
Jie Min
Numerade Educator
05:24

Problem 6

What's wrong with the following exercise? "Find the equation of the plane containing $(1,2,3),(2,3,4)$ and $(3,4,5)$."

Audrey Fong
Audrey Fong
Numerade Educator

Problem 6

Find the equation of the line through $(2,-1,-1)$ and parallel to each of the two planes $x+y=0$ and $x-y+2 z=0$. Express the equations of the line in vector and scalar parametric forms and in symmetric form.

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02:32

Problem 6

A particle traces out a curve in space, so that its position at time $t$ is

$$
\vec{r}(t)=e^{-t} \hat{\boldsymbol{\imath}}+\frac{1}{t} \hat{\boldsymbol{\jmath}}+(t-1)^2(t-3)^2 \hat{\mathbf{k}}
$$

for $t>0$.
Let the positive $z$ axis point vertically upwards, as usual. When is the particle moving upwards, and when is it moving downwards? Is it moving faster at time $t=1$ or at time $t=3$ ?

Patrick Vaughn
Patrick Vaughn
Numerade Educator
02:20

Problem 6

Draw a "contour map" of $f(x, y)=e^{-x^2+4 y^2}$, showing all types of level curves that occur.

Farnood Ensan
Farnood Ensan
Numerade Educator
View

Problem 7

Show that the set of all points $P$ that are twice as far from $(3,-2,3)$ as from $(3 / 2,1,0)$ is a sphere. Find its centre and radius.

Audrey Fleming
Audrey Fleming
Numerade Educator
02:06

Problem 7

Does the triangle with vertices $(1,2,3),(4,0,5)$ and $(3,6,4)$ have a right angle?

Cory Kuzinski
Cory Kuzinski
Numerade Educator
01:54

Problem 7

Use a projection to find the distance from the point $(-2,3)$ to the line $3 x-4 y=-4$.

Amy Jiang
Amy Jiang
Numerade Educator
01:06

Problem 7

Find the plane containing the given three points.
1. $(1,0,1),(2,4,6),(1,2,-1)$
2. $(1,-2,-3),(4,-4,4),(3,2,-3)$
3. $(1,-2,-3),(5,2,1),(-1,-4,-5)$

Raj Bala
Raj Bala
Numerade Educator
01:03

Problem 7

Let $L$ be the line given by the equations $x+y=1$ and $x+2 y+z=3$. Write a vector parametric equation for $L$.

Raj Bala
Raj Bala
Numerade Educator
00:51

Problem 7

Below is the graph of the parametrized function $\vec{r}(t)$. Let $s(t)$ be the arclength along the curve from $\vec{r}(0)$ to $\vec{r}(t)$.
Graph can't copy
Indicate on the graph $s(t+h)-s(t)$ and $\vec{r}(t+h)-\vec{r}(t)$. Are the quantities scalars or vectors?

Monica Miller
Monica Miller
Numerade Educator
07:06

Problem 7

A surface is given implicitly by

$$
x^2+y^2-z^2+2 z=0
$$

1. Sketch several level curves $z=$ constant.
2. Draw a rough sketch of the surface.

Carlos Pinilla
Carlos Pinilla
Numerade Educator
02:50

Problem 8

The pressure $p(x, y)$ at the point $(x, y)$ is at least zero and is determined by the equation $x^2-2 p x+y^2=3 p^2$. Sketch several isobars. An isobar is a curve with equation $p(x, y)=c$ for some constant $c \geq 0$.

James Kiss
James Kiss
Numerade Educator
01:10

Problem 8

Show that the area of the parallelogram determined by the vectors $\mathbf{a}$ and $\mathbf{b}$ is $|\mathbf{a} \times \mathbf{b}|$.
Figure can't copy

James Kiss
James Kiss
Numerade Educator
01:12

Problem 8

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be the vertices of a triangle. By definition, a median of a triangle is a straight line that passes through a vertex of the triangle and through the midpoint of the opposite side.
1. Find the parametric equations of the three medians.
2. Do the three medians meet at a common point? If so, which point?

Raj Bala
Raj Bala
Numerade Educator
01:54

Problem 8

Find the distance from the given point to the given plane．
1．point $(-1,2,3)$ ，plane $x+y+z=7$
2．point $(1,-4,3)$ ，plane $x-2 y+z=5$

Rukhmani Jain
Rukhmani Jain
Numerade Educator
02:06

Problem 8

1. Find a vector parametric equation for the line $x+2 y+3 z=11, x-2 y+z=-1$.
2. Find the distance from $(1,0,1)$ to the line $x+2 y+3 z=11, x-2 y+z=-1$.

WM
William Mead
Numerade Educator
02:28

Problem 8

What is the relationship between velocity and speed in a vector-valued function of time?

Shital Rijal
Shital Rijal
Numerade Educator
00:29

Problem 8

Sketch the hyperboloid $z^2=4 x^2+y^2-1$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
05:10

Problem 9

Show that the volume of the parallelepiped determined by the vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

$$
|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|
$$
Figure can't copy

Guilherme Barros
Guilherme Barros
Numerade Educator
00:42

Problem 9

Let $C$ be the circle of radius 1 centred at $(2,1)$. Find an equation for the line tangent to $C$ at the point $\left(\frac{5}{2}, 1+\frac{\sqrt{3}}{2}\right)$.

Amrita Bhasin
Amrita Bhasin
Numerade Educator

Problem 9

A plane $\Pi$ passes through the points $A=(1,1,3), B=(2,0,2)$ and $C=(2,1,0)$ in $\mathbb{R}^3$ ．
1．Find an equation for the plane $\Pi$ ．
2．Find the point $E$ in the plane $\Pi$ such that the line $L$ through $D=(6,1,2)$ and $E$ is perpendicular to $\Pi$ ．

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03:47

Problem 9

Let $L_1$ be the line passing through $(1,-2,-5)$ in the direction of $\vec{d}_1=\langle 2,3,2\rangle$. Let $L_2$ be the line passing through $(-3,4,-1)$ in the direction $\vec{d}_2=\langle 5,2,4\rangle$.
1. Find the equation of the plane $P$ that contains $L_1$ and is parallel to $L_2$.
2. Find the distance from $L_2$ to $P$.

James Kiss
James Kiss
Numerade Educator
02:21

Problem 9

Let $\vec{r}(t)$ be a vector valued function. Let $\vec{r}^{\prime}, \vec{r}^{\prime \prime}$, and $\vec{r}^{\prime \prime \prime}$ denote $\frac{\mathrm{d} \vec{r}}{\mathrm{~d} t}, \frac{\mathrm{~d}^2 \vec{r}}{\mathrm{~d} t^2}$, and $\frac{\mathrm{d}^3 \vec{r}}{\mathrm{~d} t^3}$, respectively. Express
$$
\frac{d}{d t}\left[\left(\vec{r} \times \vec{r}^{\prime}\right) \cdot \vec{r}^{\prime \prime}\right]
$$

in terms of $\vec{r}, \vec{r}^{\prime}, \vec{r}^{\prime \prime}$, and $\vec{r}^{\prime \prime \prime}$. Select the correct answer.
1. $\left(\vec{r}^{\prime} \times \vec{r}^{\prime \prime}\right) \cdot \vec{r}^{\prime \prime \prime}$
2. $\left(\vec{r}^{\prime} \times \vec{r}^{\prime \prime}\right) \cdot \vec{r}+\left(\vec{r} \times \vec{r}^{\prime}\right) \cdot \vec{r}^{\prime \prime \prime}$
3. $\left(\vec{r} \times \vec{r}^{\prime}\right) \cdot \vec{r}^{\prime \prime \prime}$
4. 0
5. None of the above.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:46

Problem 9

Describe the level surfaces of
1. $f(x, y, z)=x^2+y^2+z^2$
2. $f(x, y, z)=x+2 y+3 z$
3. $f(x, y, z)=x^2+y^2$

James Kiss
James Kiss
Numerade Educator
07:52

Problem 10

Verify by direct computation that
1. $\hat{\imath} \times \hat{\jmath}=\hat{k}, \hat{\jmath} \times \hat{k}=\hat{\imath}, \hat{k} \times \hat{\imath}=\hat{\jmath}$
2. $\mathbf{a} \cdot(\mathbf{a} \times \mathbf{b})=\mathbf{b} \cdot(\mathbf{a} \times \mathbf{b})=\mathbf{0}$

Tarandeep Singh
Tarandeep Singh
Numerade Educator
08:09

Problem 10

Let $A=(2,3,4)$ and let $L$ be the line given by the equations $x+y=1$ and $x+2 y+z=3$ ．
1．Write an equation for the plane containing $A$ and perpendicular to $L$ ．
2．Write an equation for the plane containing $A$ and $L$ ．

Matthew Markham
Matthew Markham
Numerade Educator
02:00

Problem 10

Let $L$ be a line which is parallel to the plane $2 x+y-z=5$ and perpendicular to the line $x=3-t, y=1-2 t$ and $z=3 t$.
1. Find a vector parallel to the line $L$.
2. Find parametric equations for the line $L$ if $L$ passes through a point $Q(a, b, c)$ where $a<0, b>0, c>0$, and the distances from $Q$ to the $x y$-plane, the $x z$--plane and the $y z$--plane are 2,3 and 4 respectively.

Zhumagali Shomanov
Zhumagali Shomanov
Numerade Educator
02:25

Problem 10

Find the speed of a particle with the given position function

$$
\vec{r}(t)=5 \sqrt{2} t \hat{\boldsymbol{\imath}}+e^{5 t} \hat{\boldsymbol{\jmath}}-e^{-5 t} \hat{\mathbf{k}}
$$

Select the correct answer:
1. $|\vec{v}(t)|=\left(e^{5 t}+e^{-5 t}\right)$
2. $|\vec{v}(t)|=\sqrt{10+5 e^t+5 e^{-t}}$
3. $|\vec{v}(t)|=\sqrt{10+e^{10 t}+e^{-10 t}}$
4. $|\vec{v}(t)|=5\left(e^{5 t}+e^{-5 t}\right)$
5. $|\vec{v}(t)|=5\left(e^t+e^{-t}\right)$

Amy Jiang
Amy Jiang
Numerade Educator
01:55

Problem 10

Sketch the graphs of
1. $f(x, y)=\sin x \quad 0 \leq x \leq 2 \pi, 0 \leq y \leq 1$
2. $f(x, y)=\sqrt{x^2+y^2}$
3. $f(x, y)=|x|+|y|$

Linh Vu
Linh Vu
Numerade Educator
02:03

Problem 11

Consider the following statement: "If $\mathbf{a} \neq \mathbf{0}$ and if $\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}$ then $\mathbf{b}=\mathbf{c}$." If the statment is true, prove it. If the statement is false, give a counterexample.

William Semus
William Semus
Numerade Educator
01:59

Problem 11

Consider the plane $4 x+2 y-4 z=3$ ．Find all parallel planes that are distance 2 from the above plane．Your answers should be in the following form： $4 x+2 y-4 z=C$ ．

Linh Vu
Linh Vu
Numerade Educator
02:51

Problem 11

Let $L$ be the line of intersection of the planes $x+y+z=6$ and $x-y+2 z=0$.
1. Find the points in which the line $L$ intersects the coordinate planes.
2. Find parametric equations for the line through the point $(10,11,13)$ that is perpendicular to the line $L$ and parallel to the plane $y=z$.

Linh Vu
Linh Vu
Numerade Educator

Problem 11

Find the velocity, speed and acceleration at time $t$ of the particle whose position is $\vec{r}(t)$. Describe the path of the particle.
1. $\vec{r}(t)=a \cos t \hat{\boldsymbol{i}}+a \sin t \hat{\boldsymbol{\jmath}}+c t \hat{\mathbf{k}}$
2. $\vec{r}(t)=a \cos t \sin t \hat{\boldsymbol{i}}+a \sin ^2 t \hat{\boldsymbol{\jmath}}+a \cos t \hat{\mathbf{k}}$

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03:14

Problem 11

Sketch and describe the following surfaces.
1. $4 x^2+y^2=16$
2. $x+y+2 z=4$
3. $\frac{y^2}{9}+\frac{z^2}{4}=1+\frac{x^2}{16}$
4. $y^2=x^2+z^2$
5. $\frac{x^2}{9}+\frac{y^2}{12}+\frac{z^2}{9}=1$
6. $x^2+y^2+z^2+4 x-b y+9 z-b=0 \quad$ where $b$ is a constant.
7. $\frac{x}{4}=\frac{y^2}{4}+\frac{z^2}{9}$
8. $z=x^2$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:24

Problem 12

Consider the following statement: "The vector $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ is of the form $\alpha \mathbf{b}+\beta \mathbf{c}$ for some real numbers $\alpha$ and $\beta$." If the statement is true, prove it. If the statement is false, give a counterexample.

Nick Johnson
Nick Johnson
Numerade Educator
01:04

Problem 12

Find the distance from the point $(1,2,3)$ to the plane that passes through the points $(0,1,1),(1,-1,3)$ and $(2,0,-1)$ ．

Raj Bala
Raj Bala
Numerade Educator
01:00

Problem 12

The line $L$ has vector parametric equation $\vec{r}(t)=(2+3 t) \hat{\boldsymbol{i}}+4 t \hat{\boldsymbol{\jmath}}-\hat{\mathbf{k}}$.
1. Write the symmetric equations for $L$.
2. Let $\alpha$ be the angle between the line $L$ and the plane given by the equation $x-y+2 z=0$. Find $\alpha$.

Raj Bala
Raj Bala
Numerade Educator
06:09

Problem 12

1. Let

$$
\vec{r}(t)=\left(t^2, 3, \frac{1}{3} t^3\right)
$$

Find the unit tangent vector to this parametrized curve at $t=1$, pointing in the direction of increasing $t$.
2. Find the arc length of the curve from (a) between the points ( $0,3,0$ ) and ( $1,3,-\frac{1}{3}$ ).

Susanna T.
Susanna T.
Numerade Educator
01:37

Problem 12

The surface below has circular level curves, centred along the $z$-axis. The lines given are the intersection of the surface with the right half of the $y z$-plane. Give an equation for the surface.
Figure can't copy

Linh Vu
Linh Vu
Numerade Educator

Problem 13

What geometric conclusions can you draw from $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=\langle 1,2,3\rangle$ ?

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11:00

Problem 13

Consider two planes $W_1, W_2$, and a line $M$ defined by:

$$
\begin{aligned}
W_1 & :-2 x+y+z=7 \\
W_2 & :-x+3 y+3 z=6 \\
M & : \frac{x}{2}=\frac{2 y-4}{4}=z+5
\end{aligned}
$$

1. Find a parametric equation of the line of intersection $L$ of $W_1$ and $W_2$.
2. Find the distance from $L$ to $M$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
00:52

Problem 13

Find the parametric equation for the line of intersection of the planes

$$
x+y+z=11 \quad \text { and } \quad x-y-z=13 .
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:08

Problem 13

Using Lemma 1.6.12, find the arclength of $\vec{r}(t)=\left(t, \sqrt{\frac{3}{2}} t^2, t^3\right)$ from $t=0$ to $t=1$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:39

Problem 14

What geometric conclusions can you draw from $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=0$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:16

Problem 14

Find the equation of the sphere which has the two planes $x+y+z=3, x+y+z=9$ as tangent planes if the center of the sphere is on the planes $2 x-y=0,3 x-z=0$.

Linh Vu
Linh Vu
Numerade Educator
04:52

Problem 14

1. Find a point on the $y$-axis equidistant from $(2,5,-3)$ and $(-3,6,1)$.
2. Find the equation of the plane containing the point $(1,3,1)$ and the line $\vec{r}(t)=t \hat{\boldsymbol{\imath}}+t \hat{\boldsymbol{\jmath}}+(t+2) \hat{\mathbf{k}}$.

Thomas Pauly
Thomas Pauly
Numerade Educator
02:45

Problem 14

A particle's position at time $t$ is given by $\vec{r}(t)=(t+\sin t, \cos t)^7$. What is the magnitude of the acceleration of the particle at time $t$ ?
The particle traces out a cycloid--see Question 1.6.2.4

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:03

Problem 15

Consider the three points $O=(0,0), A=(a, 0)$ and $B=(b, c)$.
1. Sketch, in a single figure,
- the triangle with vertices $O, A$ and $B$, and
- the circumscribing circle for the triangle (i.e. the circle that goes through all three vertices), and
- the vectors
- $\overrightarrow{O A}$, from $O$ to $A$,

Carson Merrill
Carson Merrill
Numerade Educator
01:03

Problem 15

Find the equation of the plane that passes through the point $(-2,0,1)$ and through the line of intersection of $2 x+3 y-z=0, x-4 y+2 z=-5$.

Carson Merrill
Carson Merrill
Numerade Educator
View

Problem 15

Let $A=(0,2,2), B=(2,2,2), C=(5,2,1)$.
1. Find the parametric equations for the line which contains $A$ and is perpendicular to the triangle $A B C$.
2. Find the equation of the set of all points $P$ such that $\overrightarrow{P A}$ is perpendicular to $\overrightarrow{P B}$. This set forms a Plane/Line/Sphere/Cone/Paraboloid/Hyperboloid (circle one) in space.
3. A light source at the origin shines on the triangle $A B C$ making a shadow on the plane $x+7 y+z=32$. (See the diagram.) Find $\tilde{A}$.
Diagram can't copy

Victor Salazar
Victor Salazar
Numerade Educator
03:13

Problem 15

A curve in $\mathbb{R}^3$ is given by the vector equation $\vec{r}(t)=\left(2 t \cos t, 2 t \sin t, \frac{t^3}{3}\right)$
1. Find the length of the curve between $t=0$ and $t=2$.
2. Find the parametric equations of the tangent line to the curve at $t=\pi$.

Stephen Hobbs
Stephen Hobbs
Numerade Educator
03:16

Problem 16

Find the equation of a sphere if one of its diameters has end points $(2,1,4)$ and $(4,3,10)$.

Howard Francis
Howard Francis
Numerade Educator
06:09

Problem 16

Find the distance from the point $\mathbf{p}$ to the plane $\mathbf{n} \cdot \mathbf{x}=c$.

Donald Albin
Donald Albin
Numerade Educator

Problem 16

Let $P, Q, R$ and $S$ be the vertices of a tetrahedron. Denote by $\vec{p}, \vec{q}, \vec{r}$ and $\vec{s}$ the vectors from the origin to $P, Q, R$ and $S$ respectively. A line is drawn from each vertex to the centroid of the opposite face, where the centroid of a triangle with vertices $\vec{a}, \vec{b}$ and $\vec{c}$ is $\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})$. Show that these four lines meet at $\frac{1}{4}(\vec{p}+\vec{q}+\vec{r}+\vec{s})$.

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03:07

Problem 16

Let $\vec{r}(t)=(3 \cos t, 3 \sin t, 4 t)$ be the position vector of a particle as a function of time $t \geq 0$.
1. Find the velocity of the particle as a function of time $t$.
2. Find the arclength of its path between $t=1$ and $t=2$.

Lucas Finney
Lucas Finney
Numerade Educator
02:50

Problem 17

Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

WZ
Wen Zheng
Numerade Educator
02:36

Problem 17

Describe the set of points equidistant from $(1,2,3)$ and $(5,2,7)$.

Aman Gupta
Aman Gupta
Numerade Educator
03:59

Problem 17

Calculate the distance between the lines $\frac{x+2}{3}=\frac{y-7}{-4}=\frac{z-2}{4}$ and $\frac{x-1}{-3}=\frac{y+2}{4}=\frac{z+1}{1}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:26

Problem 17

Consider the curve

$$
\vec{r}(t)=\frac{1}{3} \cos ^3 t \hat{\boldsymbol{\imath}}+\frac{1}{3} \sin ^3 t \hat{\boldsymbol{\jmath}}+\sin ^3 t \hat{\mathbf{k}}
$$

1. Compute the arc length of the curve from $t=0$ to $t=\frac{\pi}{2}$.
2. Compute the arc length of the curve from $t=0$ to $t=\pi$.

Jordan Fowlds
Jordan Fowlds
Numerade Educator
05:46

Problem 18

Compute the areas of the parallelograms determined by the following vectors.
1. $\langle-3,1\rangle,\langle 4,3\rangle$
2. $\langle 4,2\rangle,\langle 6,8\rangle$

Carlos Pinilla
Carlos Pinilla
Numerade Educator
03:36

Problem 18

Describe the set of points equidistant from $\mathbf{a}$ and $\mathbf{b}$.

Mengchun Cai
Mengchun Cai
Numerade Educator
02:42

Problem 18

Let $\vec{r}(t)=\left(\frac{1}{3} t^3, \frac{1}{2} t^2, \frac{1}{2} t\right), t \geq 0$. Compute $s(t)$, the arclength of the curve at time $t$.

Linh Vu
Linh Vu
Numerade Educator
01:43

Problem 19

Consider the plane $W$, defined by:

$$
W:-x+3 y+3 z=6
$$

Find the area of the parallelogram on $W$ defined by $0 \leq x \leq 3,0 \leq y \leq 2$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:57

Problem 19

Consider a point $P(5,-10,2)$ and the triangle with vertices $A(0,1,1), B(1,0,1)$ and $C(1,3,0)$.
1. Compute the area of the triangle $A B C$.
2. Find the distance from the point $P$ to the plane containing the triangle.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:35

Problem 19

Find the arc length of the curve $\vec{r}(t)=\left(t^m, t^m, t^{3 m / 2}\right)$ for $0 \leq a \leq t \leq b$, and where $m>0$. Express your result in terms of $m, a$, and $b$.

Joseph Liao
Joseph Liao
Numerade Educator
06:37

Problem 20

Compute the volumes of the parallelepipeds determined by the following vectors.
1. $\langle 4,1,-1\rangle,\langle-1,5,2\rangle,\langle 1,1,6\rangle$
2. $\langle-2,1,2\rangle,\langle 3,1,2\rangle,\langle 0,2,5\rangle$

Antonio Fonte
Antonio Fonte
Numerade Educator
02:08

Problem 20

Consider the sphere given by

$$
(x-1)^2+(y-2)^2+(z+1)^2=2
$$

Suppose that you are at the point $(2,2,0)$ on $S$, and you plan to follow the shortest path on $S$ to $(2,1,-1)$. Express your initial direction as a cross product.

Steven Clarke
Steven Clarke
Numerade Educator
03:07

Problem 20

If a particle has constant mass $m$, position $\vec{r}$, and is moving with velocity $\vec{v}$, then its angular momentum is $\mathbf{L}=m(\vec{r} \times \vec{v})$. For a particle with mass $m=1$ and position function $\vec{r}=(\sin t, \cos t, t)$, find $\backslash(\operatorname{left} \mid$ frac $\{\mathrm{mathrm}\{\mathrm{d}\} \backslash \operatorname{textbf}\{\mathrm{L}\}$ $\{$ mathrm $\{\mathrm{d}\} t\} \backslash \operatorname{right}(|\operatorname{text}\{\}|$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:39

Problem 21

Compute the dot product of the vectors $\mathbf{a}$ and $\mathbf{b}$. Find the angle between them.
1. $\mathbf{a}=\langle 1,2\rangle, \mathbf{b}=\langle-2,3\rangle$
2. $\mathbf{a}=\langle-1,1\rangle, \mathbf{b}=\langle 1,1\rangle$
3. $\mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 2,2\rangle$
4. $\mathbf{a}=\langle 1,2,1\rangle, \mathbf{b}=\langle-1,1,1\rangle$
5. $\mathbf{a}=\langle-1,2,3\rangle, \mathbf{b}=\langle 3,0,1\rangle$

Lucas Finney
Lucas Finney
Numerade Educator
02:59

Problem 21

Consider the space curve $\Gamma$ whose vector equation is

$$
\vec{r}(t)=t \sin (\pi t) \hat{\boldsymbol{\imath}}+t \cos (\pi t) \hat{\boldsymbol{\jmath}}+t^2 \hat{\mathbf{k}} \quad 0 \leq t<\infty
$$

This curve starts from the origin and eventually reaches the ellipsoid $E$ whose equation is $2 x^2+2 y^2+z^2=24$.
1. Determine the coordinates of the point $P$ where $\Gamma$ intersects $E$.
2. Find the tangent vector of $\Gamma$ at the point $P$.
3. Does $\Gamma$ intersect $E$ at right angles? Why or why not?

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:44

Problem 22

Determine the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ if
1. $\mathbf{a}=\langle 1,2\rangle, \mathbf{b}=\langle 3,4\rangle$
2. $\mathbf{a}=\langle 2,1,4\rangle, \mathbf{b}=\langle 4,-2,1\rangle$
3. $\mathbf{a}=\langle 1,-2,1\rangle, \mathbf{b}=\langle 3,1,0\rangle$

Lucas Finney
Lucas Finney
Numerade Educator
01:04

Problem 22

Suppose a particle in 3-dimensional space travels with position vector $\vec{r}(t)$, which satisfies $\vec{r}^{\prime \prime}(t)=-\vec{r}(t)$. Show that the "energy" $|\vec{r}(t)|^2+\left|\vec{r}^{\prime}(t)\right|^2$ is constant (that is, independent of $t$ ).

Raj Bala
Raj Bala
Numerade Educator
01:06

Problem 23

Determine all values of $y$ for which the given vectors are perpendicular.
1. $\langle 2,4\rangle,\langle 2, y\rangle$
2. $\langle 4,-1\rangle,\left\langle y, y^2\right\rangle$
3. $\langle 3,1,1\rangle,\left\langle 2,5 y, y^2\right\rangle$

Jin Yan Yeo
Jin Yan Yeo
Numerade Educator

Problem 23

A particle moves along the curve $\backslash \mathrm{cC}$ of intersection of the surfaces $z^2=12 y$ and $18 x=y z$ in the upward direction. When the particle is at $(1,3,6)$ its velocity $\vec{v}$ and acceleration $\vec{a}$ are given by

$$
\vec{v}=6 \hat{\boldsymbol{\imath}}+12 \hat{\boldsymbol{\jmath}}+12 \hat{\mathbf{k}} \quad \vec{a}=27 \hat{\boldsymbol{\imath}}+30 \hat{\boldsymbol{\jmath}}+6 \hat{\mathbf{k}}
$$

1. Write a vector parametric equation for $\backslash \mathrm{cC}$ using $u=\frac{z}{6}$ as a parameter.
2. Find the length of $\backslash \mathrm{cC}$ from $(0,0,0)$ to $(1,3,6)$.
3. If $u=u(t)$ is the parameter value for the particle's position at time $t$, find $\frac{d u}{d t}$ when the particle is at $(1,3,6)$.
4. Find $\frac{\mathrm{d}^2 u}{\mathrm{~d} t^2}$ when the particle is at $(1,3,6)$.

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01:00

Problem 24

Let $\mathbf{u}=-2 \hat{\imath}+5 \hat{\jmath}$ and $\mathbf{v}=\alpha \hat{\imath}-2 \hat{\jmath}$. Find $\alpha$ so that
1. $\mathbf{u} \perp \mathbf{v}$
2. $\mathbf{u} \| \mathbf{v}$
3. The angle between $\mathbf{u}$ and $\mathbf{v}$ is $60^{\circ}$.

Samantha Lincroft
Samantha Lincroft
Numerade Educator
05:46

Problem 24

A particle of mass $m=1$ has position $\vec{r}_0=\frac{1}{2} \hat{\mathbf{k}}$ and velocity $\vec{v}_0=\frac{\pi^2}{2} \hat{\boldsymbol{\imath}}$ at time 0 . It moves under a force

$$
\mathbf{F}(t)=-3 t \hat{\boldsymbol{\imath}}+\sin t \hat{\boldsymbol{\jmath}}+2 e^{2 t} \hat{\mathbf{k}}
$$

1. Determine the position $\vec{r}(t)$ of the particle depending on $t$.
2. At what time after time $t=0$ does the particle cross the plane $x=0$ for the first time?
3. What is the velocity of the particle when it crosses the plane $x=0$ in part (b)?

Kevin Hayakawa
Kevin Hayakawa
University of California - Los Angeles
01:07

Problem 25

Define $\mathbf{a}=\langle 1,2,3\rangle$ and $\mathbf{b}=\langle 4,10,6\rangle$.
1. Find the component of $\mathbf{b}$ in the direction $\mathbf{a}$.
2. Find the projection of $\mathbf{b}$ on $\mathbf{a}$.
3. Find the projection of $\mathbf{b}$ perpendicular to $\mathbf{a}$.

Tyler Moulton
Tyler Moulton
Numerade Educator
04:40

Problem 25

Let $C$ be the curve of intersection of the surfaces $y=x^2$ and $z=\frac{2}{3} x^3$. A particle moves along $C$ with constant speed such that $\frac{\mathrm{d} x}{\mathrm{~d} t}>0$. The particle is at $(0,0,0)$ at time $t=0$ and is at $(3,9,18)$ at time $t=\frac{7}{2}$.
1. Find the length of the part of $C$ between $(0,0,0)$ and $(3,9,18)$.
2. Find the constant speed of the particle.
3. Find the velocity of the particle when it is at $\left(1,1, \frac{2}{3}\right)$.
4. Find the acceleration of the particle when it is at $\left(1,1, \frac{2}{3}\right)$.

Carson Merrill
Carson Merrill
Numerade Educator
01:31

Problem 26

Compute $\langle 1,2,3\rangle \times\langle 4,5,6\rangle$.

Anas Venkitta
Anas Venkitta
Numerade Educator
03:26

Problem 26

A camera mounted to a pole can swivel around in a full circle. It is tracking an object whose position at time $t$ seconds is $x(t)$ metres east of the pole, and $y(t)$ metres north of the pole.
In order to always be pointing directly at the object, how fast should the camera be programmed to rotate at time $t$ ? (Give your answer in terms of $x(t)$ and $y(t)$ and their derivatives, in the units rad/sec.)

Ahmad Reda
Ahmad Reda
Numerade Educator
01:03

Problem 27

Calculate the following cross products.
1. $\langle 1,-5,2\rangle \times\langle-2,1,5\rangle$
2. $\langle 2,-3,-5\rangle \times\langle 4,-2,7\rangle$
3. $\langle-1,0,1\rangle \times\langle 0,4,5\rangle$

Tyler Moulton
Tyler Moulton
Numerade Educator
03:23

Problem 27

A projectile falling under the influence of gravity and slowed by air resistance proportional to its speed has position satisfying

$$
\frac{d^2 \vec{r}}{d t^2}=-g \hat{\mathbf{k}}-\alpha \frac{d \vec{r}}{d t}
$$

where $\alpha$ is a positive constant. If $\vec{r}=\vec{r}_0$ and $\frac{d \vec{r}}{d t}=\vec{v}_0$ at time $t=0$, find $\vec{r}(t)$. (Hint: Define $\mathbf{u}(t)=e^{\alpha t} \frac{d \vec{r}}{d t}(t)$ and substitute $\frac{d \vec{r}}{d t}(t)=e^{-\alpha t} \mathbf{u}(t)$ into the given differential equation to find a differential equation for $\mathbf{u}$.)

Nick Johnson
Nick Johnson
Numerade Educator

Problem 28

Let $\mathbf{p}=\langle-1,4,2\rangle, \mathbf{q}=\langle 3,1,-1\rangle, \mathbf{r}=\langle 2,-3,-1\rangle$. Check, by direct computation, that
1. $\mathbf{p} \times \mathbf{p}=\mathbf{0}$
2. $\mathbf{p} \times \mathbf{q}=-\mathbf{q} \times \mathbf{p}$
3. $\mathbf{p} \times(3 \mathbf{r})=3(\mathbf{p} \times \mathbf{r})$
4. $\mathbf{p} \times(\mathbf{q}+\mathbf{r})=\mathbf{p} \times \mathbf{q}+\mathbf{p} \times \mathbf{r}$
5. $\mathbf{p} \times(\mathbf{q} \times \mathbf{r}) \neq(\mathbf{p} \times \mathbf{q}) \times \mathbf{r}$

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02:03

Problem 28

At time $t=0$ a particle has position and velocity vectors $\vec{r}(0)=\langle-1,0,0\rangle$ and $\vec{v}(0)=\langle 0,-1,1\rangle$. At time $t$, the particle has acceleration vector

$$
\vec{a}(t)=\langle\cos t, \sin t, 0\rangle
$$

1. Find the position of the particle after $t$ seconds.
2. Show that the velocity and acceleration of the particle are always perpendicular for every $t$.
3. Find the equation of the tangent line to the particle's path at $t=-\pi / 2$.
4. True or False: None of the lines tangent to the path of the particle pass through $(0,0,0)$. Justify your answer.

Averell Hause
Averell Hause
Carnegie Mellon University

Problem 29

Calculate the area of the triangle with vertices $(0,0,0),(1,2,3)$ and $(3,2,1)$.

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02:28

Problem 29

The position of a particle at time $t$ (measured in seconds s ) is given by

$$
\vec{r}(t)=t \cos \left(\frac{\pi t}{2}\right) \hat{\boldsymbol{\imath}}+t \sin \left(\frac{\pi t}{2}\right) \hat{\boldsymbol{\jmath}}+t \hat{\mathbf{k}}
$$

1. Show that the path of the particle lies on the cone $z^2=x^2+y^2$.
2. Find the velocity vector and the speed at time $t$.
3. Suppose that at time $t=1 \mathrm{~s}$ the particle flies off the path on a line $L$ in the direction tangent to the path. Find the equation of the line $L$.
4. How long does it take for the particle to hit the plane $x=-1$ after it started moving along the straight line $L$ ?

Amy Jiang
Amy Jiang
Numerade Educator

Problem 30

A particle $P$ of unit mass whose position in space at time $t$ is $\mathbf{r}(t)$ has angular momentum $L(t)=\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)$. If $\mathbf{r}^{\prime \prime}(t)=\rho(t) \mathbf{r}(t)$ for a scalar function $\rho$, show that $L$ is constant, i.e. does not change with time. Here ' denotes $\frac{d}{d t}$.

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10:42

Problem 30

1. The curve $\vec{r}_1(t)=\left\langle 1+t, t^2, t^3\right\rangle$ and $\vec{r}_2(t)=\langle\cos t, \sin t, t\rangle$ intersect at the point $P(1,0,0)$. Find the angle of intersection between the curves at the point $P$.
2. Find the distance between the line of intersection of the planes $x+y-z=4$ and $2 x-z=4$ and the line $\vec{r}(t)=\langle t,-1+2 t, 1+3 t\rangle$.

Bobby Barnes
Bobby Barnes
University of North Texas
01:45

Problem 31

Show that the diagonals of a parallelogram bisect each other.

Benjamin Chaback
Benjamin Chaback
Numerade Educator
View

Problem 32

Consider a cube such that each side has length $s$. Name, in order, the four vertices on the bottom of the cube $A, B, C, D$ and the corresponding four vertices on the top of the cube $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$.
1. Show that all edges of the tetrahedron $A^{\prime} C^{\prime} B D$ have the same length.
2. Let $E$ be the center of the cube. Find the angle between $E A$ and $E C$.

Victor Salazar
Victor Salazar
Numerade Educator
02:08

Problem 33

Find the angle between the diagonal of a cube and the diagonal of one of its faces.

WZ
Wen Zheng
Numerade Educator
02:36

Problem 34

Consider a skier who is sliding without friction on the hill $y=h(x)$ in a two dimensional world. The skier is subject to two forces. One is gravity. The other acts perpendicularly to the hill. The second force automatically adjusts its magnitude so as to prevent the skier from burrowing into the hill. Suppose that the skier became airborne at some ( $x_0, y_0$ ) with $y_0=h\left(x_0\right)$. How fast was the skier going?

Adnan Gill
Adnan Gill
Numerade Educator
00:51

Problem 35

A marble is placed on the plane $a x+b y+c z=d$. The coordinate system has been chosen so that the positive $z$-axis points straight up. The coefficient $c$ is nonzero and the coefficients $a$ and $b$ are not both zero. In which direction does the marble roll? Why were the conditions " $c \neq 0$ " and " $a, b$ not both zero" imposed?

Joseph Liao
Joseph Liao
Numerade Educator
View

Problem 36

Show that $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 37

Show that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$.

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04:11

Problem 38

Derive a formula for $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})$ that involves dot but not cross products.

Manisha Sarker
Manisha Sarker
Numerade Educator
05:56

Problem 39

A prism has the six vertices

$$
\begin{array}{ll}
A=(1,0,0) & A^{\prime}=(5,0,1) \\
B=(0,3,0) & B^{\prime}=(4,3,1) \\
C=(0,0,4) & C^{\prime}=(4,0,5)
\end{array}
$$

1. Verify that three of the faces are parallelograms. Are they rectangular?
2. Find the length of $A A^{\prime}$.
3. Find the area of the triangle $A B C$.
4. Find the volume of the prism.

Barsha Rana
Barsha Rana
Numerade Educator
03:36

Problem 40

(Three dimensional Pythagorean Theorem) A solid body in space with exactly four vertices is called a tetrahedron. Let $A, B$, $C$ and $D$ be the areas of the four faces of a tetrahedron. Suppose that the three edges meeting at the vertex opposite the face of area $D$ are perpendicular to each other. Show that $D^2=A^2+B^2+C^2$.
Figure can't copy

Hannah Wilds
Hannah Wilds
Numerade Educator

Problem 41

(Three dimensional law of cosines) Let $A, B, C$ and $D$ be the areas of the four faces of a tetrahedron. Let $\alpha$ be the angle between the faces with areas $B$ and $C, \beta$ be the angle between the faces with areas $A$ and $C$ and $\gamma$ be the angle between the faces with areas $A$ and $B$. (By definition, the angle between two faces is the angle between the normal vectors to the faces.) Show that

$$
D^2=A^2+B^2+C^2-2 B C \cos \alpha-2 A C \cos \beta-2 A B \cos \gamma
$$

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