Calculus

Laura Taalman, Peter Kohn

Chapter 10

Vectors - all with Video Answers

Educators

Section 1

Cartesian Coordinates

Problem 1

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: The distance between two distinct points can be zero.
(b) True or False: In the Cartesian plane the equation $x=5$ represents a line.
(c) True or False: In three-dimensional space the equation $x=5$ represents a line.
(d) True or False: In four-dimensional space the equation $x=5$ represents a plane.
(e) True or False: The equation
$$
x^{2}+y^{2}+z^{2}+2 x-4 y-10 z+50=0
$$
represents a sphere with center (-1,2,5) .
(f) True or False: If a sphere in $\mathbb{R}^{3}$ has its center in the first octant and is tangent to each of the coordinate planes, then its center is at the point $(c, c, c)$ for some constant $c$.
(g) True or False: When two distinct spheres intersect, they intersect in either a point or a circle.
(h) True or False: Three noncollinear points in $\mathbb{R}^{3}$ determine a unique plane.

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A plane parallel to the $y z$ -plane.
(b) A sphere tangent to the $x y$ -plane.
(c) A sphere tangent to all three coordinate planes.

Lucas Finney

Numerade Educator

Problem 3

Consider the equations $y=5$ and $x=-3$.
(a) What do these equations represent in a twodimensional system?
(b) What do these equations represent in a threedimensional system?

Lucas Finney

Numerade Educator

Problem 4

Consider the equation $x+2 y=4$
(a) What does this equation represent in a twodimensional system?
(b) What does this equation represent in a threedimensional system?

Lucas Finney

Numerade Educator

Problem 5

What are the coordinates of the vertices of a cube with side length $2,$ whose center is at the origin, and whose faces are parallel to the coordinate planes?

Lucas Finney

Numerade Educator

Problem 6

The sides of a $2 \times 3 \times 4$ rectangular solid are parallel to the coordinate planes. The coordinates of four of its vertices are $(1,-2,3),(-1,-2,-1),(-1,1,3),$ and (1,-2,3) What are the coordinates of the other four vertices?

Lucas Finney

Numerade Educator

Problem 7

What is the definition of a sphere?

Lucas Finney

Numerade Educator

Problem 8

What is the definition of a cylinder? What is the directrix? What is a ruling?

Lucas Finney

Numerade Educator

Problem 9

Consider the equation $x^{2}+y^{2}=4$
(a) What does this equation represent in a twodimensional system?
(b) What does this equation represent in a threedimensional system?

Lucas Finney

Numerade Educator

Problem 10

Consider the equation $z=y^{2}$.
(a) What does this equation represent in the $y z$ -plane?
(b) What does this equation represent in a threedimensional system?

Lucas Finney

Numerade Educator

Problem 11

What point is symmetric about the origin to the point (5,-6,7)$?$

Lucas Finney

Numerade Educator

Problem 12

What point is symmetric to the point (-1,3,6) with respect to the $x y$ -plane?

Lucas Finney

Numerade Educator

Problem 13

What point is symmetric to the point (3,-7,-4) with respect to the plane $z=1 ?$

Lucas Finney

Numerade Educator

Problem 14

Find all of the $x, y, z$ labelings of the axes in the diagram that follows. Determine which of your labelings are righthanded systems and which are left-handed systems.

Sarah Parrigin

Numerade Educator

Problem 14

Show that exchanging two of the axes labels on a righthanded system creates a left-handed system. What does exchanging two pairs of axes labels do (for example, exchanging $x$ and $y$ and then exchanging the "new" $y$ and $z$ )?

Lucas Finney

Numerade Educator

Problem 16

Sketch the point (2,3,4) in a three-dimensional Cartesian coordinate system, and then answer the following questions:
(a) Do the coordinates (2,3,4) represent a unique point in a three-dimensional Cartesian coordinate system?
(b) Are there any other coordinates $(a, b, c)$ that would have the same location as (2,3,4) in your graph?
(c) How far from the $x y$ -plane is the point (2,3,4) ?
(d) How far from the $x z$ -plane is the point (2,3,4) ?
(e) How far from the $y z$ -plane is the point (2,3,4) ?
(f) How far from the $x$ -axis is the point (2,3,4) ?
(g) How far from the origin is the point (2,3,4) ?
(h) How far from the point (1,-2,3) is the point (2,3,4)$?$
(i) What is the equation of the sphere with center (2,3,4) and passing through the point (1,-2,3)$?$
(j) What is the equation of the plane parallel to the $x y$ -plane and that contains the point (2,3,4)$?$
(k) What is the equation of the plane parallel to the $x z$ -plane and that contains the point (2,3,4) ?
(1) What is the equation of the plane parallel to the $y z$ -plane and that contains the point (2,3,4) ?

Carson Merrill

Numerade Educator

Problem 17

How does the Pythagorean theorem generalize to higher dimensions? In particular, how would you compute the distance between two points in four-dimensional space? Five-dimensional space? $n$ -dimensional space?

Lucas Finney

Numerade Educator

Problem 18

The points in the first octant satisfy the inequalities $x>0$, $y>0, z>0 .$ For each of the other seven octants, find a set of inequalities that describes the points in the octant. Use the inequalities to label the octants of the following right-handed coordinate system:

Lucas Finney

Numerade Educator

Problem 19

Use inequalities to describe each of the following sets, and sketch the regions:
(a) The region in the first octant above the plane $z=3$.
(b) The region inside the sphere with center (1,2,4) and radius 5
(c) The region inside the right circular cylinder with directrix $x^{2}+y^{2}-4 x+6 y-12=0$

Sarah Parrigin

Numerade Educator

Problem 20

Sketch the right circular cylinders:
$$
x^{2}+y^{2}=1, x^{2}+z^{2}=4, \text { and } y^{2}+z^{2}=9
$$

WZ

Wen Zheng

Numerade Educator

Problem 21

Find the distance between the given pair of points.
$$
(-2,3) \text { and }(5,-6)
$$

Vikash Ranjan

Numerade Educator

Problem 22

Find the distance between the given pair of points.
$$
(4,0) \text { and }(-5,12)
$$

Vikash Ranjan

Numerade Educator

Problem 23

Find the distance between the given pair of points.
$$
(1,4,7) \text { and }(-2,3,5)
$$

Vikash Ranjan

Numerade Educator

Problem 24

Find the distance between the given pair of points.
$$
(3,0,-1) \text { and }(2,-8,0)
$$

Vikash Ranjan

Numerade Educator

Problem 25

Find the distance between the given pair of points.
$$
(-1,4,-3) \text { and }(-4,3,1)
$$

Vikash Ranjan

Numerade Educator

Problem 26

Find the distance between the given pair of points.
$$
(4,5,8,-2) \text { and }(-1,3,-3,6)
$$

Vikash Ranjan

Numerade Educator

Problem 27

Find the distance between the given pair of points.
$$
(-1,3,5,2,0) \text { and }(0,6,1,-2,3)
$$

Vikash Ranjan

Numerade Educator

Problem 28

Find the distance between the given pair of points.
$$
(0,0, \ldots, 0) \text { and }(1,1, \ldots, 1) \text { in } n \text { -space }
$$

Vikash Ranjan

Numerade Educator

Problem 29

Find an equation of a sphere with the specified characteristics.
center (3,-2,5) and radius 5

Vikash Ranjan

Numerade Educator

Problem 30

Find an equation of a sphere with the specified characteristics.
center (4,-2,-3) and radius 3

Vikash Ranjan

Numerade Educator

Problem 31

Find an equation of a sphere with the specified characteristics.
center (2,5,-7) and tangent to the $x y$ -plane

Vikash Ranjan

Numerade Educator

Problem 32

Find an equation of a sphere with the specified characteristics.
center (2,5,-7) and tangent to the $y z$ -plane

Vikash Ranjan

Numerade Educator

Problem 33

Find an equation of a sphere with the specified characteristics.
center (2,5,-7) and containing the origin

Vikash Ranjan

Numerade Educator

Problem 34

Find an equation of a sphere with the specified characteristics.
containing the point (1,4,7) and whose center is (-2,3,5)

Vikash Ranjan

Numerade Educator

Problem 35

Find an equation of a sphere with the specified characteristics.
containing the point (3,0,-1) and whose center is (2,-8,0)

Vikash Ranjan

Numerade Educator

Problem 36

Use the midpoint formula to find the equations of the spheres.
the sphere in which the segment with endpoints (3,-2,6) and (5,6,4) is a diameter

Vikash Ranjan

Numerade Educator

Problem 37

Use the midpoint formula to find the equations of the spheres.
the sphere in which the segment with endpoints (6,-1,4) and (-3,3,1) is a diameter

Vikash Ranjan

Numerade Educator

Problem 38

Find the center and radius of the sphere with the given equation.
$$
x^{2}+y^{2}+z^{2}-2 x+6 y-8 z+17=0
$$

Vikash Ranjan

Numerade Educator

Problem 39

Find the center and radius of the sphere with the given equation.
$$
x^{2}+y^{2}+z^{2}+3 y-5 z+3=0
$$

Vikash Ranjan

Numerade Educator

Problem 40

Find the equations of the intersections of the sphere
$$
x^{2}+y^{2}+z^{2}-2 x+6 y-8 z+17=0
$$
with each of the coordinate planes.

Lucas Finney

Numerade Educator

Problem 41

Graph the quadric surfaces given by the equations.
$$
x^{2}=\frac{y^{2}}{9}+\frac{z^{2}}{9}
$$

Vikash Ranjan

Numerade Educator

Problem 42

Graph the quadric surfaces given by the equations.
$$
z^{2}=\frac{x^{2}}{9}+\frac{y^{2}}{25}
$$

Vikash Ranjan

Numerade Educator

Problem 43

Graph the quadric surfaces given by the equations.
$$
x^{2}+y^{2}+1=z^{2}
$$

Vikash Ranjan

Numerade Educator

Problem 44

Graph the quadric surfaces given by the equations.
$$
x^{2}+y^{2}-z^{2}=1
$$

Vikash Ranjan

Numerade Educator

Problem 45

Graph the quadric surfaces given by the equations.
$$
z=\frac{x^{2}}{9}+\frac{y^{2}}{25}
$$

Vikash Ranjan

Numerade Educator

Problem 46

Graph the quadric surfaces given by the equations.
$$
9 x^{2}+16 y^{2}+16 z^{2}=144
$$

Vikash Ranjan

Numerade Educator

Problem 47

Graph the quadric surfaces given by the equations.
$$
z=x^{2}-y^{2}
$$

Vikash Ranjan

Numerade Educator

Problem 48

Graph the quadric surfaces given by the equations.
$$
z=x^{2}+y^{2}
$$

Vikash Ranjan

Numerade Educator

Problem 49

A circle is inscribed in a square so that each side of the square is tangent to the circle. A smaller circle is inscribed in the square so that this circle is tangent to two sides of the square and is tangent to the larger circle, as shown in the following figure:
What is the ratio of the radius of the smaller circle to the radius of the larger circle?

Georgiann Andersen

Georgiann Andersen

Numerade Educator

Problem 50

A sphere is inscribed in a cube so that each face of the cube is tangent to the sphere. A smaller sphere is inscribed in the cube so that this sphere is tangent to three sides of the cube and is tangent to the larger sphere, as shown in the following figure:
What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? (Hint: Try Exercise 49 first.)

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 51

Show that the triangle with vertices (5,4,-1),(3,6,-1) and (3,4,1) is equilateral.

Vikash Ranjan

Numerade Educator

Problem 52

Show that the triangle with vertices (1,2,-2),(-3,2,-6) and (-3,6,-2) is equilateral.

Vikash Ranjan

Numerade Educator

Problem 53

Show that the points $(1,5,0),(3,8,6),$ and (7,-7,4) are the vertices of a right triangle and find its area.

Nick Johnson

Numerade Educator

Problem 54

A regular tetrahedron is a solid with four faces in which each face is an equilateral triangle of the same size. You are asked some basic questions about regular tetrahedra.
(a) Show that the three points $(1,0,0),(0,1,0),$ and (0,0,1) are the vertices of an equilateral triangle.
(b) Determine the two values of $a$ so that the four points $(1,0,0),(0,1,0),(0,0,1),$ and $(a, a, a)$ are the vertices of a regular tetrahedron.

Yaqub Khan

Numerade Educator

Problem 55

A regular tetrahedron is a solid with four faces in which each face is an equilateral triangle of the same size. You are asked some basic questions about regular tetrahedra.
Find the equations of the spheres that circumscribe the two tetrahedra you determined in Exercise 54 (b). (Hint:
The center of the sphere is the point $\left(x_{0}, y_{0}, z_{0}\right),$ where $x_{0}$ is the mean of the $x$ -coordinates of the four vertices of the tetrahedron, etc.)

Yaqub Khan

Numerade Educator

Problem 56

Show that the six points $(1,0,0),(-1,0,0),(0,1,0),$ $(0,-1,0),(0,0,1),$ and (0,0,-1) form the vertices of a regular octahedron. (A regular octahedron is an eight-sided solid in which each face is an equilateral triangle of the same size, and in which four triangles come together at each vertex.)

Khushbu Rani

Numerade Educator

Problem 57

Find the volume of the octahedron given in Exercise $56 .$ (Hint: Recall that the volume of a pyramid is $\frac{1}{3} \cdot$ height. area of base.)

SE

Steven Ellsmoor

Numerade Educator

Problem 58

Ian is doing a high traverse. One morning he looks at the map and notes that if he considers his camp to be at the origin, then his objective is at $(5.9,3.3,-0.37) .$ All distances are in miles.
(a) How far away is his objective, as the crow flies?
(b) In order to reach his objective, Ian has to go over a high pass that lies at (4.2,4.4,0.15) relative to his camp. Find a more realistic estimate of how far he has to go to his objective than that from part (a).

Matthew Baker

Numerade Educator

Problem 59

Annie is in a kayak in the middle of a channel between Orcas Island and Blakely Island, two of the San Juan Islands in Washington State. She knows from readings she has taken in the past that if she considers the town of Deer Harbor to be at the origin, then the summit of Constitution Peak is at $(7.9,4.0),$ while the summit of Blakely Peak is at (9.3,-3.4) . All distances are given in miles. She pulls out her compass and finds that the summit of Constitution Peak is 15 degrees east of north from her, while the summit of Blakely Peak is at 100 degrees from north. How far is Annie from Deer Harbor?

Km Neeraj

Numerade Educator

Problem 60

The Subaru reflecting telescope on Mauna Kea, Hawaii, has a mirror in the shape of a circular paraboloid with a diameter of 8.3 meters and a focal length of 15 meters. While there are some tricks to how that focal length plays out in practice, if we put the center of the telescope at the origin, pointed straight up, then the effective focus would be at $(0,0, p),$ where $p$ satisfies $4 p z=x^{2}+y^{2},$ with all distances given in meters. How high above the $x y$ -plane is the edge of the telescope?

Victor Salazar

Numerade Educator

Problem 61

The Hyper Potato Chip Company makes potato chips in the shape of hyperbolic paraboloids. Each chip satisfies the equation $z=\frac{x^{2}}{a^{2}}-\frac{y^{2}}{a^{2}}$ when the center of the chip is placed at the origin. The height of each Hyper chip is 0.16 inch, the length is 2.6 inches, and the width is 1.6 inches. Find the parameter $a$ and write the equation of a Hyper chip.

Carson Merrill

Numerade Educator

Problem 62

Prove that the midpoint of the line segment connecting the point $\left(x_{1}, y_{1}, z_{1}\right)$ to the point $\left(x_{2}, y_{2}, z_{2}\right)$ is $\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$

Sarah Parrigin

Numerade Educator

Problem 63

Given any edge $\mathcal{E}$ of a tetrahedron, there is exactly one edge $\mathcal{E}^{\prime}$ that does not share a face with $\mathcal{E}$. We will call $\mathcal{E}$ and $\mathcal{E}^{\prime}$ opposite edges of the tetrahedron. Prove that the line segments connecting the midpoints of the opposite edges of a regular tetrahedron bisect each other.

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 64

Prove that the midpoint of the line segment connecting the points $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $\left(y_{1}, y_{2}, \ldots, y_{n}\right)$ in $\mathbb{R}^{n}$
is $\left(\frac{x_{1}+y_{1}}{2}, \frac{x_{2}+y_{2}}{2}, \ldots, \frac{x_{n}+y_{n}}{2}\right)$

Sarah Parrigin

Numerade Educator