Deck 13: Vectors and the Geometry of Space

Question 1

Describe the given set of points with a single equation or with a pair of equations.
The circle of radius 6 centered at the point (6, -4, 36) and lying in a plane parallel to the xy-plane

Answer 1

B

Question 2

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

A) All points between z = 10 and z = 11 in the x-y plane
B) The slab between the planes z = 10 and z = 11 (including planes)
C) No set of points satisfies the given relation
D) The line from z = 10 to z = 11

Answer 2

B

Question 3

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

A) The square with corners at (0, 3, 3), (0, 3, 4), (0, 4, 3), and (0, 4, 4)
B) The infinitely long square prism parallel to the x-axis
C) The line between the points (0, 3, 3) and (0, 4, 4)
D) The cube located in the first quadrant and with sides 3 units in length

Answer 3

B

Question 4

Describe the given set of points with a single equation or with a pair of equations.
The circle in which the plane through the point (-8, 12, ) perpendicular to the y-axis meets the sphere of radius 15 centered at the origin.

Answer 4

A

Question 5

Describe the given set of points with a single equation or with a pair of equations.
The circle of radius 2 centered at the point (10, -2, 2) and lying in a plane perpendicular to the x-axis

Question 6

Write one or more inequalities that describe the set of points.
The slab bounded by the planes x = and x = 5 (planes included)

Question 7

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

Question 8

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

A) No set of points satisfy the given relations.
B) All points on the lower hemisphere centered at (10, 2, 2)
C) All points within the lower hemisphere centered at (10, 2, 2)
D) All points outside the lower hemisphere centered at (10, 2, 2)

Question 9

Describe the given set of points with a single equation or with a pair of equations.
The circle in which the plane through the point (5, -8, -2) perpendicular to the x-axis meets the sphere of radius 13 centered at the origin.

Question 10

Describe the given set of points with a single equation or with a pair of equations.
The plane through the point (-7, -3, 2) and parallel to the xy-plane

A) x + y + 2 = 0
B) x + y = -10
C) z = 2
D) x = -7 and y = -3

Question 11

Describe the given set of points with a single equation or with a pair of equations.
The plane perpendicular to the y-axis and passing through the point (2, -1, 7)

A) x - 1 + z = 0
B) y = -1
C) 2x + 7z = 0 and y = -1
D) x = 2 and z = 7

Question 12

Describe the given set of points with a single equation or with a pair of equations.
The set of points equidistant from the points (0, 0, -6) and (0, 0, 4)

A) x + y = -1
B) z = -1
C) z > -6 and z < 4
D) x + y = 0 and -6 < z < 4

Question 13

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

Question 14

Describe the given set of points with a single equation or with a pair of equations.
The plane through the point (1, -4, -5) and parallel to the yz-plane

A) x = 1
B) 1 + y + z = 0
C) -4y - 5z = 0 and x = 1
D) y = -4 and z = -5

Question 15

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

Question 16

Describe the given set of points with a single equation or with a pair of equations.
The set of points equidistant from the points (-1, 0, 0) and (2, 0, 0)

A) y + z = 0.5
B) y + z = 0 and -1 < x < 2
C) x = 0.5
D) x > -1 and x < 2

Question 17

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

A) All points outside the cylinder with radius 3
B) All points outside the sphere of radius 3
C) All points in space
D) All points on the surface of the cylinder with radius 3

Question 18

Describe the given set of points with a single equation or with a pair of equations.
The plane through the point (-1, -2, -5) and perpendicular to the x-axis

A) y = -2 and z = -5
B) -1 + y + z = 0
C) x = -1
D) y + z = -7

Question 19

Write one or more inequalities that describe the set of points.
The rectangular solid in the first octant bounded by the planes x = 5, x = 8, y = 7, y = 9, z = 6 and z = 7 (planes excluded)

A) The given planes do not form a rectangular solid
B) x < y < z
C) 5 < x < 8; 7 < y < 9; 6 < z < 7
D) x < 5, x > 8; y < 7, y > 9; z < 6, z > 7

Question 20

Give a geometric description of the set of points whose coordinates satisfy the given conditions.
x = , z =

A) All points in the x-z plane
B) The point (9, 2)
C) The line through the point (9, 2, 0) and parallel to the z-axis
D) The line through the point (9, 0, 2) and parallel to the y-axis

Question 21

Write one or more inequalities that describe the set of points.
The exterior of the sphere of radius 5 centered at the point (-2, -3, -5)

Question 22

Question 23

Write one or more inequalities that describe the set of points.

Question 24

Find an equation for the sphere with the given center and radius.
Center (-5, 7, 0), radius = 8

Question 25

Find an equation for the sphere with the given center and radius.
Center (0, 0, 10), radius = 8

Question 26

Find the center and radius of the sphere.

Question 27

Find an equation for the sphere with the given center and radius.
Center (-2, 0, 0), radius = 9

Question 28

Question 29

Find the center and radius of the sphere.

A) C(0, 10, -1), a = 64
B) C(0, -10, 1), a = 64
C) C(0, 10, -1), a = 8
D) C(0, -10, 1), a = 8

Question 30

Find the center and radius of the sphere.

A) C(4, 1, 4), a = 6
B) C(4, 1, -4), a = 6
C) C(4, 1, -4), a = 36
D) C(-4, -1, 4), a = 6

Question 31

Write one or more inequalities that describe the set of points.
The closed region bounded by the spheres of radius 2 and 7, both centered at the origin, and the planes x = 3 and x = 5

Question 32

Question 33

A) 15
B) 18
C) 14
D) 10

Question 34

Find an equation for the sphere with the given center and radius.
Center (0, -8, -1), radius = 10

Question 35

Find the center and radius of the sphere.

A) C(9, 5, 3), a = 100
B) C(-9, -5, -3), a = 10
C) C(-9, -5, -3), a = 100
D) C(9, 5, 3), a = 10

Question 36

Solve the problem.
Find a formula for the distance from the point P(x, y, z) to the z-axis.

Question 37

Solve the problem.
Find a formula for the distance from the point P(x, y, z) to the xz plane.

Question 38

Write one or more inequalities that describe the set of points.
The half-space consisting of the points on and behind the yz-plane

Question 39

Find the center and radius of the sphere.

Question 40

Find the center and radius of the sphere.

Question 41

Find the component form of the specified vector.
The vector from the point A(8, 1) to the origin

Question 42

A) v = 5i - 7j - 3k
B) v = 5i - 7j + 3k
C) v = 5i + 7j - 3k
D) v = 5i + 7j + 3k

Question 43

Find the magnitude.

Question 44

Find the indicated vector.

Question 45

Find the magnitude.

Question 46

Find the indicated vector.

Question 47

Find the indicated vector.

Question 48

Find the indicated vector.

Question 49

Find the component form of the specified vector.

Question 50

Find the component form of the specified vector.

Question 51

Find the indicated vector.

Question 52

Solve the problem.
Find the perimeter of the triangle with vertices A(2, 2, 6), B(2, -2, 6), and C(3, 5, 7).

Question 53

Solve the problem.
Show that the point P( , -5, ) is equidistant from the points A( , -7, ) and B( , -3, ).

Question 54

Find the magnitude.

Question 55

Find the indicated vector.

Question 56

Find the indicated vector.

Question 57

Find the indicated vector.

Question 58

Find the indicated vector.

Question 59

Find the indicated vector.

Question 60

A) v = -2i + 3j - 4k
B) v = -2i - 3j + 4k
C) v = -2i + 3j + 4k
D) v = 2i - 3j - 4k

Question 61

Express the vector as a product of its length and direction.
-5i + 5j + 5k

Question 62

Express the vector as a product of its length and direction.
7j

Question 63

Express the vector as a product of its length and direction.

Question 64

Question 65

Express the vector as a product of its length and direction.

Question 66

Express the vector as a product of its length and direction.
5i + 10j + 10k

Question 67

Question 68

Solve the problem.
Find a vector of magnitude 11 in the direction of v = 5i - 12k.

Question 69

Express the vector as a product of its length and direction.

Question 70

Question 71

Express the vector as a product of its length and direction.

Question 72

Question 73

Express the vector as a product of its length and direction.

Question 74

Express the vector as a product of its length and direction.

Question 75

Question 76

Solve the problem.

Question 77

A) v = -8i + 12j - 5k
B) v = 22i + 7j - 5k
C) v = 7i + 7j - 5k
D) v = -8i + 7j - 5k

Question 78

Question 79

Question 80

Express the vector as a product of its length and direction.