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Chapter 12

Vectors and the Geometry of Space

Educators

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ag
BT
+ 11 more educators

Problem 1

Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

YZ
Yiming Z.
Numerade Educator

Problem 2

Sketch the points $ (1, 5, 3) $, $ (0, 2, -3) $, $ (-3, 0, 2) $, and $ (2, -2, -1) $ on a single set of coordinate axes.

ag
Alan G.
Numerade Educator

Problem 3

Which of the points $ A (-4, 0, -1) $, $ B (3, 1, -5) $, and $ C (2, 4, 6) $ is closest to the $ yz $-plane? Which point lies in the $ xz $-plane?

YZ
Yiming Z.
Numerade Educator

Problem 4

What are the projections of the point $ (2, 3, 5) $ on the $ xy $-, $ yz $-, and $ xz $- planes? Draw a rectangular box with the origin and $ (2, 3, 5) $ as opposite vertices and with its faces parallel to the
coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

BT
Brandon T.
Numerade Educator

Problem 5

What does the equation $ x = 4 $ represent in $ \mathbb{R}^2 $? What does it represent in $ \mathbb{R}^3 $? Illustrate with sketches.

YZ
Yiming Z.
Numerade Educator

Problem 6

What does the equation $ y = 3 $ represent in $ \mathbb{R}^3 $? What does $ z = 5 $ represent? What does the pair of equations $ y = 3 $, $ z = 5 $ represent? In other words, describe the set of points $ (x, y, z) $ such that $ y = 3 $ and $ z = 5 $. Illustrate with a sketch.

YZ
Yiming Z.
Numerade Educator

Problem 7

Describe and sketch the surface in $ \mathbb{R}^3 $ represented by the equation $ x + y = 2 $.

Bobby B.
University of North Texas

Problem 8

Describe and sketch the surface $ \mathbb{R}^3 $ represented by the equation $ x^2 + z^2 = 9 $.

Jacquelyn T.
Numerade Educator

Problem 9

Find the lengths of the sides of the triangle $ PQR $. Is it a right triangle? Is it an isosceles triangle?

$ P (3, -2, -3) $ , $ Q (7, 0, 1) $ , $ R (1, 2, 1) $

ag
Alan G.
Numerade Educator

Problem 10

Find the lengths of the sides of the triangle $ PQR $. Is it a right triangle? Is it an isosceles triangle?

$ P (2, -1, 0) $ , $ Q (4, 1, 1) $ , $ R (4, -5, 4) $

Jacquelyn T.
Numerade Educator

Problem 11

Determine whether the points lie on a straight line.

(a) $ A (2, 4, 2) $ , $ B (3, 7, -2) $, $ C (1, 3, 3) $
(b) $ D (0, -5, 5) $ , $ E (1, -2, 4) $, $ F (3, 4, 2) $

DB
Daniel B.
Numerade Educator

Problem 12

Find the distance from $ (4, -2, 6) $ to each of the following.

(a) The $ xy $-plane (b) The $ yz $-plane
(c) The $ xz $-plane (d) The $ x $-axis
(e) The $ y $-axis (f) The $ z $-axis

Adam D.
Numerade Educator

Problem 13

Find an equation of the sphere with center $ (-3, 2, 5) $ and radius 4. What is the intersection of this sphere with the $ yz $-plane?

ag
Alan G.
Numerade Educator

Problem 14

Find an equation of the sphere with center $ (2, -6, 4) $ and radius 5. Describe its intersection with each of the coordinate planes.

Jacquelyn T.
Numerade Educator

Problem 15

Find an equation of the sphere that passes through the point $ (4, 3, -1) $ and has center $ (3, 8, 1) $.

ag
Alan G.
Numerade Educator

Problem 16

Find an equation of the sphere that passes through the origin and whose center is $ (1, 2, 3) $.

YZ
Yiming Z.
Numerade Educator

Problem 17

Show that the equation represents a sphere, and find its center and radius.

$ x^2 + y^2 + z^2 - 2x - 4y + 8z = 15 $

ag
Alan G.
Numerade Educator

Problem 18

Show that the equation represents a sphere, and find its center and radius.

$ x^2 + y^2 + z^2 + 8x - 6y + 2z + 17 = 0 $

ag
Alan G.
Numerade Educator

Problem 19

Show that the equation represents a sphere, and find its center and radius.

$ 2x^2 + 2y^2 + 2z^2 = 8x - 24z + 1 $

ag
Alan G.
Numerade Educator

Problem 20

Show that the equation represents a sphere, and find its center and radius.

$ 3x^2 + 3y^2 + 3z^2 = 10 + 6y + 12z $

Bobby B.
University of North Texas

Problem 21

(a) Prove that the midpoint of the line segment from $ P_1 (x_1, y_1, z_1) $ to $ P_2 (x_2, y_2, z_2) $ is
$$ \left(\frac{x_1 + x_2}{2} , \frac{y_1+ y_2}{2} , \frac{z_1+ z_2}{2} \right) $$
(b) Find the lengths of the medians of the triangle with vertices $ A (1, 2, 3) $, $ B (-2, 0, 5) $, $ C (4, 1, 5) $. (A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.)

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Problem 22

Find an equation of a sphere if one of its diameters has endpoints $ (5, 4, 3) $ and $ (1, 6, -9) $.

YZ
Yiming Z.
Numerade Educator

Problem 23

Find equations of the spheres with center $ (2, -3, 6) $ that touch (a) the $ xy $-plane, (b) the $ yz $-plane, (c) the $ xz $-plane.

Sirat S.
Numerade Educator

Problem 24

Find an equation of the largest sphere with center $ (5, 4, 9) $ that is contained in the first octant.

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Problem 25

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x = 5 $

YZ
Yiming Z.
Numerade Educator

Problem 26

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ y = -2 $

ag
Alan G.
Numerade Educator

Problem 27

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ y < 8 $

ag
Alan G.
Numerade Educator

Problem 28

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ z \ge -1 $

ag
Alan G.
Numerade Educator

Problem 29

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ 0 \le z \le 6 $

ag
Alan G.
Numerade Educator

Problem 30

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ y^2 = 4 $

ag
Alan G.
Numerade Educator

Problem 31

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + y^2 = 4 $, $ z = -1 $

ag
Alan G.
Numerade Educator

Problem 32

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + y^2 = 4 $

YZ
Yiming Z.
Numerade Educator

Problem 33

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + y^2 + z^2 = 4 $

YZ
Yiming Z.
Numerade Educator

Problem 34

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + y^2 + z^2 \le 4 $

ag
Alan G.
Numerade Educator

Problem 35

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ 1 \le x^2 + y^2 + z^2 \le 5 $

ag
Alan G.
Numerade Educator

Problem 36

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x = z $

Megan S.
Numerade Educator

Problem 37

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + z^2 \le 9 $

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Problem 38

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + y^2 + z^2 > 2z $

Isaac B.
Numerade Educator

Problem 39

Write inequalities to describe the region.
The region between the yz-plane and the vertical plane $ x = 5 $

Pawan Y.
Numerade Educator

Problem 40

The solid cylinder that lies on or below the plane $ z = 8 $ and on or above the disk in the $ xy $-plane with center the origin and radius 2.

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Problem 41

The region consisting of all points between (but not on) the spheres of radius $ r $ and $ R $ centered at the origin, where $ r < R $.

CP
Connor P.
Numerade Educator

Problem 42

The solid upper hemisphere of the sphere of radius 2 centered at the origin

Hannah M.
Numerade Educator

Problem 43

The figure shows a line $ L_1 $ in space and a second line $ L_2 $, which is the projection of $ L_1 $ onto the $ xy $-plane. (In other words, the points on $ L_2 $ are directly beneath, or above, the points on $ L_1 $.)
(a) Find the coordinates of the point $ P $ on the line $ L_1 $.
(b) Locate on the diagram the points $ A $, $ B $, and $ C $, where the line $ L_1 $ intersects the $ xy $-plane, the $ yz $-plane, and the $ xz $-plane, respectively.

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Problem 44

Consider the points $ P $ such that the distance from $ P $ to $ A (-1, 5, 3) $ is twice the distance from $ P $ to $ B (6, 2, -2) $. Show that the set of all such points is a sphere, and find its center and radius.

Grant W.
Numerade Educator

Problem 45

Find an equation of the set of all points equidistant from the points $ A (-1, 5, 3) $ and $ B (6, 2, -2) $. Describe the set.

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Problem 46

Find the volume of the solid that lies inside both of the spheres
$$ x^2 + y^2 + z^2 + 4x -2y + 4z + 5 = 0 $$
and
$$ x^2 + y^2 + z^2 = 4 $$

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Problem 47

Find the distance between the spheres $ x^2 + y^2 + z^2 = 4 $ and $ x^2 + y^2 + z^2 = 4x + 4y + 4z - 11 $.

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Problem 48

Describe and sketch a solid with the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles triangle.

Janielle M.
Numerade Educator