Deck 11: Vectors and Coordinate Geometry in 3-Space

Question 1

Describe the surface with equation x² + y² + z² - 6x + 2y - 12z = 75.

A) sphere with centre at (3, -1, 6) and radius
B) sphere with centre at (-3, 1, -6) and radius
C) sphere with centre at (3, 1, 6) and radius 11
D) sphere with centre at (3, -1, 6) and radius 11
E) sphere with centre at (3, 1, 6) and radius

Answer 1

sphere with centre at (3, -1, 6) and radius 11

Question 2

Find the distance between the points (x, y, z) and (2, 0, -3).

A)
B) + +
C)
D) + +
E)

Answer 2

Question 3

Find the distance between the points (2, 4, 7) and the (1, 5, 10).

A) 3
B)
C)
D)
E) 11

Answer 3

Question 4

Find the distance between (2, -1, -2) and the origin.

A)
B)
C) 3
D) 4
E) 9

Question 5

Find the distance between (-2, 3, 1) and (4, 1, -3).

A)
B) 2
C)
D)
E) 2

Question 6

Find the equation of the sphere with radius 7 and centre (-1, 12, 9).

A) + + = 49
B) + + = 7
C) + + = 7
D) + + = 49
E) + + =

Question 7

Describe the 3-space graph of the equation x² + y² = 0.

A) the xy-plane
B) the z-axis
C) a circle
D) a circular cylinder
E) a parabolic cylinder

Question 8

Describe the 3-space graph of the equation x² + z² = 4.

A) a circle of radius 2 in the xz-plane having centre at the origin
B) a circular cylinder of radius 2 with central axis along the y-axis
C) a sphere of radius 2 having centre at the origin
D) a disk of radius 2 in the xz-plane having centre at the origin.
E) a circular cylinder of radius 4 with central axis along the x-axis

Question 9

Describe the 3-space graph of the equation z² = y² + x².

A) a surface of revolution obtained by rotating the parabola z = x² in the xz-plane about the z-axis
B) a circular cone with vertex at the origin, axis along the z-axis, and semi-vertical angle
C) a circular cylinder with axis along the z-axis radius z
D) a circle having radius z in the xy-plane
E) a surface of revolution obtained by rotating the parabola z = y² in the yz-plane about the z-axis

Question 10

Describe the set of points in 3-space defined by the equations x² + y² + z² - 8z = 9, x = 3.

A) a circle of radius 3 lying in the plane x = 3 and having centre at (3, 0, 4)
B) a circle of radius 3 lying in the plane x = 3 and having centre at (3, 0, -4)
C) a circle of radius 4 lying in the plane x = 3 and having centre at (3, 0, -4)
D) a circle of radius 4 lying in the plane x = 3 and having centre at (3, 0, 4)
E) a sphere of radius 5 having centre at (3, 0, 4)

Question 11

If line L passes through point (1, 2, 3) and is perpendicular to the xy-plane, what are the coordinates of the points on the line that are at a distance 7 from the point P(3, -1, 5)?

A) (1, 2, 1) and (1, 2, 9)
B) (1, 2, 0) and (1, 2, 10)
C) (1, 2, -1) and (1, 2, 11)
D) (1, 2, -2) and (1, -2, 12)
E) (1, 2, -4) and (1, 2, 10)

Question 12

Describe the intersection of the graphs of x² + z² = 4 and y = 2.

A) circle of radius 4 with centre at (0, 2, 0) in the plane z = 2
B) circle of radius 2 with centre at (0, 2, 0) in the plane y = 2
C) circle of radius 2 with centre at (0, -2, 0) in the plane y = -2
D) circle of radius 2 with centre at (1, 2, 1) in the plane y = 2
E) circle of radius with centre at (0, 2, 0) in the plane y = 2

Question 13

Describe the intersection of the graphs of y = x and y = 5 in 3-space.

A) line through the point (5, 5, 0) and perpendicular to the xy-plane
B) line through the points (5, 5, 5) and (0, 0, 0)
C) line through the point (0, 5, 0) and perpendicular to the xy-plane
D) line through the points (5, 5, 0) and (0, 0, 0)
E) line through the points (0, 0, 0) and (5, 5, 0)

Question 14

Describe the intersection of the graphs of y = x and y² + z² = 9.

A) the ellipse in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis
B) the circle in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis
C) the hyperbola in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis
D) the parabola in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis
E) the ellipse in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the y-axis

Question 15

Describe the intersection of the graphs of y = 3 and x² + y² = z².

A) the parabola in which the vertical plane perpendicular to the y-axis and the point (0, 3, 0) intersects the right circular cone with axis along the x-axis and semi-vertical angle
B) the circle in which the vertical plane perpendicular to the y-axis and the point (0, 3, 0) intersects the right circular cone with axis along the x-axis and semi-vertical angle
C) the hyperbola in which the vertical plane perpendicular to the y-axis and the point (0, 3, 0) intersects the right circular cone with axis along the x-axis and semi-vertical angle
D) the ellipse in which the vertical plane perpendicular to the y-axis and the point (0, 3, 0) intersects the right circular cone with axis along the x-axis and semi-vertical angle
E) none of the above

Question 16

Find an equation of the sphere with centre in the xz-plane and passing through the points(0, 8, 0), (4, 6, 2), and (0, 12, 4).

A) x² + y² + z² - 14x + 24z = 257
B) x² + y² + z² + 14x - 24z = 257
C) x² + y² + z² - 14x + 24z = 64
D) x² + y² + z² + 14x - 24z = 64
E) x² + y² + z² - 14x - 24z = 257

Question 17

Find the point on the y-axis equidistant from (2, 5, -3) and (-3, 6, 1).

A) (0, -5, 0)
B) (0, 4, 0)
C) (0, 3, 2)
D) (0, -2, 0)
E) (0, -4, 0)

Question 18

Find an equation describing all points that are equidistant from the points A(-3, 0, 4) and B(2, 1, 5). What does this equation describe geometrically?

A) 10x + 2y + 2z = 5, the plane that right bisects the line segment AB
B) 10x - 2y + 2z = 5, the plane that right bisects the line segment AB
C) 10x - 2y + 18z = 5, the plane that right bisects the line segment AB
D) 10x + 2y + 18z = 5, the plane that right bisects the line segment AB
E) 10x + 2y + 2z = 5, the plane that contains the points A and B

Question 19

When does the equation x² + y² + z² + Ax + By + Cz + D = 0 represent a sphere?

A) if and only if A² + B² + C² + D > 0
B) if and only if A² + B² - C² > D
C) if and only if A² - B² + C² > 0
D) if and only if A² + B² + C² > 4D
E) if and only if A = 0, B = 0, C = 0, and D < 0

Question 20

There are fewer than two points on the graph of x² + y² + z² + 2z + 2 = 0.

Question 21

If v = -i + 7j and w = 4i - 4j, then find v + w.

A) 4i + 11j
B) 3i + 11j
C) 3i + 3j
D) 4i + 3j
E) 5i + 3j

Question 22

Find the components of the unit vector in the same direction as v = .

A)
B)
C)
D)
E)

Question 23

If v is a vector in the xy-plane, |v| = 7 , and v makes an angle of 3 $\pi$ /4 with the positive direction of the x-axis, then what are the components of v?

A)
B)
C)
D)
E)

Question 24

Find the length of the vector PQ where P = (1, -2, 4) and Q = (3, 4, 3).

A)
B) 2
C)
D)
E)

Question 25

Find the angle $\theta$ between the vectors i + 2j - 3k and -i + 2j + k.

A) $\pi$
B)
C)
D)
E)

Question 26

Find the angle $\theta$ between the vectors i + 2j + 3k and 2i - 3j - k.

A)
B)
C)
D)
E)

Question 27

Find a value c for which A = 2i - j + 4k and B = i + cj + 8k will be perpendicular.

A) c = -18
B) c = -12
C) c = -34
D) c = 34
E) 0

Question 28

Find the vector of unit length in the direction opposite to that of v = 4i + 7j - 4k.

A) i + j - k
B) i + j - k
C) - i - j + k
D) - i - j + k
E) - i - j + k

Question 29

Find the vector that makes equal acute angles with the positive coordinate axes and has length 2.

A) (i + j + k)
B) 2(i + j + k)
C) (i + j + k)
D) (i + j + k)
E) (i + j + k)

Question 30

Let u and v be non-zero vectors in 2 or 3-space. Find a simplified expression for .

A)
B) + 2 +
C) + 2 +
D) + 2 +
E) + 2u.v +

Question 31

Find the scalar and vector projections of 2i -5j + k in the direction of 3i - 4k.

A) scalar 2, vector (3i - 4k)
B) scalar , vector (3i - 4k)
C) scalar , vector (3i - 4k)
D) scalar , vector (3i - 4k)
E) scalar , vector (3i - 4k)

Question 32

A vector in 100-dimensional Euclidean space R¹⁰⁰ makes equal acute angles with the positive directions of the 100 coordinate axes. Approximately what is that angle?

A) 68.71°
B) 84.26°
C) 75.44°
D) 87.18°
E) 45.00°

Question 33

Given u = i - 2j + k and v = 3i + j - 2k, find each of the following:
(a) u x v, (b) v x u, and (c) v x v.

Question 34

Given u = i - 2j + k and v = 3i + j - 2k, find each of the following:(a) u x v, (b) v x u, and (c) v x v.

A) (a) 3i + 5j + 7k, (b) -3i - 5j - 7k, (c) 0
B) (a) 3i - 5j + 7k, (b) -3i + 5j - 7k, (c) -4i +12j
C) (a) 3i + 5j - 5k, (b) -3i - 5j + 5k, (c) 12j
D) (a) 3i - 5j - 5k, (b) -3i + 5j + 5k, (c) -4 i + 6k
E) (a) 3i + 5j + 7k, (b) -3i - 5j - 5k, (c) 0

Question 35

Calculate u × v where u = 2i + j - k and v = -3i + 4j + k.

A) 5i + j - 11k
B) 5i - j + 11k
C) 5i + j + 11k
D) -5i + j + 11k
E) -5i + j - 11k

Question 36

If u = and v = , evaluate u x v.

A)
B)
C)
D)
E)

Question 37

Given u = 2i - j + 2k and v = i - j + 4k, find each of the following:(a) u × v, (b) v × u, and (c) v × v.

A) (a) -2i - 6j - k, (b) 2i + 6j + k, (c) 0
B) (a) -2i - 6j - k, (b) -2i - 6j - k, (c) -4i +8j -4k
C) (a) 2i + 6j + k, (b) -2i - 6j - k, (c) -4i +8j -4k
D) (a) -2i - 6j + k, (b) 2i + 6j - k, (c) 0
E) (a) -2i + 6j - k, (b) 2i - 6j + k, (c) 0

Question 38

Find the area of a triangle that has vertices (4, 3, 6), (-2, 0, 8), (1, 5, 0).

A) square units
B) 49 square units
C) square units
D) square units
E) 28 square units

Question 39

Find the two unit vectors orthogonal to both a = 3j + 2k and b = - i - 2k.

A) - i - j + k and i + j - k
B) k and -k
C) - i + j and i - j
D) - i + j + k and i - j - k
E) -6i - 2j + 3k and 6i + 2j - 3k

Question 40

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (3v - w).

A) 41i + 11j + 28k
B) 41i - 11j + 28k
C) -41i - 11j + 28k
D) -41i + 11j + 28k
E) 41i - 11j - 28k

Question 41

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (v × w).

A) -13i + 19j + 5k
B) -14i + 20j + 4k
C) -13i - 19j - 5k
D) 14i - 20j - 4k
E) 13i - 19j - 5k

Question 42

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u . (v × w).

A) 63
B) 64
C) -64
D) 62
E) -63

Question 43

If u, v, and w are vectors in 3-space and u × v = u × w, then v = w.

Question 44

u × (v × w) = (u × v) × w for all vectors u, v, and w in 3-space.

Question 45

Find the volume of a parallelepiped spanned by vectors from the origin to the three points (1, 1, -3), (-1, 3, -1), and (3, 5, 7).

A) 70 cubic units
B) 72 cubic units
C) 68 cubic units
D) 76 cubic units
E) 11 cubic units

Question 46

Find the volume of the tetrahedron spanned by the vectors u = , v = , and w = .

A) 2 cubic units
B) 3 cubic units
C) 4 cubic units
D) 5 cubic units
E) 1 cubic unit

Question 47

A force F of magnitude 6 N acts in the direction of the vector i + 2j- 2k and is applied at the point . (Distances are in centimetres.) What is the magnitude of the torque of F about the point ?

A) 14 N . cm
B) 5 N . cm
C) 10 N . cm
D) 15 N . cm
E) 30 N . cm

Question 48

Given three vectors u, v, and w, with v and w not 0 or parallel, find the vector projection of u in the plane containing the origin and the points represented by the position vectors v and w.

A) - (v × w)
B) (v × w) - u
C) u - (v × w)
D) (v × w)
E) none of the above

Question 49

If u × v = 0 (the zero vector), then either u = 0 or v = 0.

Question 50

Write standard form equations for the line through the point (1, 2, -6) and parallel to the vector .

A) = =
B) = =
C) = =
D) = =
E) = = -

Question 51

The planes x + 2y - 4z = 10 and -2x - 4y - 8z = 11 are parallel.

Question 52

Find the cosine of the angle $\theta$ between the planes 3x + 3y - 6z = 12 and 7x - 5y + 2z = -8.

A) ±
B) ±
C) ±
D) ±
E) ±

Question 53

Find the equation for the plane that passes through the point (1, 2, 3) and is normal to the vector joining (1, 3, 2) and (2, 3, 1).

A) x - z + 2 = 0
B) x + z +2 = 0
C) 2x - z +1 = 0
D) x - 2z -1 = 0
E) x + z -2= 0

Question 54

Find the equation of a plane containing the point (2, -4, 3) and the line = = z + 2.

A) 19x + 14y - z = -21
B) 19x - 14y - z = 91
C) 19x - 14y + z = 97
D) 19x + 14y + z = -15
E) 19x + 14y - 2z = 91

Question 55

Find the equation of the straight line passing through the point (1, 2, -3) and is perpendicular to the plane 3x - 7y +4z - 17 = 0.

A) r = (1 + 3t) i + (2 - 7t) j + (- 3 + 4t) k, t (- $\infty$ , $\infty$ )
B) 3x - 7y + 4z - 23 = 0
C) r = (3 + t) i + (- 7 + 2t) j + (4 - 3t) k, t 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (- $\infty$ , $\infty$ )
D) x + y + z = 0
E)

Question 56

Find the acute angle between the planes with equations
x + y - z = 3 and x + y + z = 5.

A)
B)
C)
D)
E)

Question 57

Find the distance from the point P(1, -1, 2) to the line connecting the point (3, 1, 4) to (1, -3, 0).

A) units
B) units
C) units
D) units
E) units

Question 58

Find parametric equations of the straight line containing the point (2, 1, -4) and is parallel to the line of intersection of the planes 3x +2y - z = 0 and x + y + z = -9.

Question 59

Find the distance between the lines = = and = = .

A) units
B) 3 units
C) 2 units
D) 3 units
E) 1 unit

Question 60

Find the equation of a plane that contains the line = = and is parallel to the plane 2x - 3y + 2z = 0.

A) 2x - 3y + 2z = 15
B) 2x - 3y + 2z = -15
C) 2x - 3y + 2z = 12
D) 2x - 3y + 2z = -12
E) 2x + 3y + 2z = 15

Question 61

Find the equation of a plane that contains the lines= = and - == z - 2.

A) 53x + 51y - 37z = 67
B) 53x - 47y + 5z = -731
C) 53x - 47y +57 = 11
D) 53x + 51y - 37z = -67
E) 53x - 47y - 5z = 11

Question 62

For what values of the constants k and c does the line = = lie in the plane x - y + 2z = c?

A) k = -10, c = 1
B) k = 10, c = 0
C) k = 10, c = -1
D) k = -8, c = 2
E) k = -2, c = 0

Question 63

For what value of the constant k will the vectors , , and be coplanar?

A) k = 8
B) k = -7
C) k = 6
D) k = -5
E) k = 0

Question 64

The distance from the point (2, -1, -2) to the plane 6x + 2y - 3z + a = 0 is 2 units. Find a.

A) 2
B) -2 or -30
C) -14
D) -10 or -22
E) -14, -18

Question 65

A plane in 3-space is uniquely determined by any three different points that lie on it.

Question 66

Consider the straight line L: = =
(i) Find the point on the line L closest to the point P (-2, -1, 3).
(ii) Find the shortest distance from the point P to the line L.

Question 67

Find the coordinates of the point where the line that passes through the point (0, -3, 8) and is parallel to the line given by x = 10 + 3t, y = 12t, and z = -3 - t intersects the xz-plane.

A)
B)
C)
D)
E)

Question 68

Determine the equation of the plane that contains the points P = (1, -2, 0), Q = (3, 1, 4), and R = (0, -1, 2).

A) 2x - 8y + 5z = 18
B) 2x + 8y + 5z = -14
C) 2x + 8y - 5z = -14
D) 2x - 8y - 5z = 18
E) 2x + 8y + 5z = 18

Question 69

Describe the graph of 9x² - y² + 16z² = 144.

A) a hyperboloid of one sheet
B) a hyperboloid of two sheets
C) a hyperbolic paraboloid
D) an elliptic paraboloid
E) an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 70

Describe the graph of 25x² - y² - z² = 25.

A) a hyperboloid of one sheet
B) a hyperboloid of two sheets
C) a hyperbolic paraboloid
D) an elliptic paraboloid
E) an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 71

Which of the following equations is an equation of a circular cone?

A) z² = 1 + 4x² + 4y²
B) z = x² + y² + z²
C) z = 1 + x² + y²
D) x² + z² = 1
E) z² = 4(x² + y²)

Question 72

Which of the following equations is an equation of a hyperboloid of one sheet?

A) x² + y² - z² = 0
B) x² - y² - z² + 1= 0
C) x² + y² - z² + 1= 0
D) z² = x² - y²
E) 2x² - y + 3z² = 1

Question 73

Describe the graph of x² + 4z² = 2y.

A) a hyperboloid of one sheet
B) a hyperboloid of two sheets
C) a hyperbolic paraboloid
D) an elliptic paraboloid
E) an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 74

Describe the graph of 4x² - y² = 2x + 3y.

A) a hyperboloid of one sheet
B) a hyperboloid of two sheets
C) a hyperbolic paraboloid
D) an elliptic paraboloid
E) an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 75

Describe the graph of x² + 4y² + 16z² = 2x - 8y.

A) a hyperboloid of one sheet
B) a hyperboloid of two sheets
C) a hyperbolic paraboloid
D) an elliptic paraboloid
E) an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 76

Which of the following is an equation of a hyperboloid of two sheets?

A) x² + y² - z² = 0
B) x² - y² - z² = -1
C) z² = 1 - x² - y²
D) x² + y² + 1 = z²
E) z = - x² - y²

Question 77

Describe the set of points in 3-space satisfying z² = x² + y² and z = x + y.

A) two straight lines
B) one straight line
C) an ellipse
D) a parabola
E) a hyperbola
F) none of the above

Question 78

Describe the set of points in 3-space satisfying z² = x² + y² and z = 2x.

A) two straight lines
B) one straight line
C) an ellipse
D) a parabola
E) a hyperbola
F) none of the above

Question 79

Describe the set of points in 3-space satisfying z² = x² + y² and z = 1 + y.

A) two straight lines
B) one straight line
C) an ellipse (or a circle)
D) a parabola
E) a hyperbola
F) none of the above

Question 80

Describe the set of points in 3-space satisfying x² + 2y² + 3z² = 4 and z = x + y.

A) two straight lines
B) one straight line
C) an ellipse
D) a parabola
E) a hyperbola
F) none of the above