Deck 12: Vectors and the Geometry of Space

Question 1

Classify the surface. $23 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 12 y + 12 z = 0$

A) A hyperboloid of one sheet with center $( - 8,0,6 )$ and axis parallel to the z-axis.
B) A cone with axis parallel to the z-axis and vertex $( 0,6,6 )$ .
C) A circular paraboloid with vertex $( 0,8,6 )$ and axis the z-axis.

Answer 1

A cone with axis parallel to the z-axis and vertex $( 0,6,6 )$ .

Question 2

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is $\text { three }$ times the distance from P to the yz-plane.

A) $9 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$
B) $3 x ^ { 2 } = y ^ { 2 } + 3 z$
C) $x ^ { 2 } - y ^ { 2 } - z ^ { 2 } = 3$
D) $3 x ^ { 2 } = y ^ { 2 } - 3 z ^ { 2 }$
E) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$

Answer 2

$9 x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$

Question 3

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the x-axis. Find the equation of the ellipsoid.

A) $16 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$
B) $4 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 16$
C) $4 x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 16$
D) $4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$
E) $$4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$$

Answer 3

$$4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$$

Question 4

Find an equation of the plane with x-intercept $= 11$ , y-intercept $= 5$ and z-intercept $$= 5$$ .

A) $$\frac { x } { 11 } + \frac { y } { 5 } + \frac { z } { 5 } = 1$$
B) $11 x + 5 y + 5 z = 0$
C) $\frac { 5 x } { 5 } + 11 y + z = 0$
D) $\frac { x } { 11 } + \frac { y } { 5 } + \frac { z } { 5 } = 1$
E) $11 x + 5 y + 5 z = 1$

Question 5

Reduce the equation to one of the standard forms. $z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$

A) $x ^ { 2 } + y ^ { 2 } - \frac { z ^ { 2 } } { 4 } = 1$
B) $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 6 } - \frac { z ^ { 2 } } { 24 } = 1$
C) $x ^ { 2 } + \frac { y ^ { 2 } } { 2 } - \frac { z ^ { 2 } } { 4 } = 1$
D) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } - z ^ { 2 } = 1$
E) $( x - 4 ) ^ { 2 } + \frac { y ^ { 2 } } { \left( \frac { 1 } { 4 } \right) ^ { 2 } } + z ^ { 2 } = 1$

Question 6

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the y-axis.

A) $y ^ { 2 } + 3 x ^ { 2 } = z$
B) $z ^ { 2 } + 3 x ^ { 2 } = y$
C) $z ^ { 2 } + 3 x ^ { 2 } + y ^ { 2 } = 1$
D) $z ^ { 2 } - 3 x ^ { 2 } - y ^ { 2 } = 1$
E) $z ^ { 2 } - 3 x ^ { 2 } = y$

Question 7

Which of the given lines is parallel to the line $x = 8 + t , y = t , z = - 7 - 10 t ?$

A) $x = - 5 + 3 t , y = 10 + t , z = - 2 - t$
B) $x + 8 = y - 2 = - 9 - z$
C) $x = - 5 t , y = 10 t , z = - 1 - t$
D) $x = - 5 + t , y = 10 + t , z = - 1 - t$
E) $r = \langle - 6 , - 10,0 \} + t ( 2,2 , - 20 )$

Question 8

Find an equation of the set of all points equidistant from the points $A ( 3 , - 4 , - 5 )$ and $B ( - 4,2 , - 10 )$

A) $$7 x - 6 y + 5 z = - 35$$
B) $7 x - 6 y + 5 z = - 35$
C) $7 x - 6 y + 5 z = 35$
D) $7 x - 6 y = 0$
E) $- 7 x - 6 y + 5 z = - 35$

Question 9

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

A) $z ^ { 2 } - x ^ { 2 } = - 8 y$
B) $x ^ { 2 } + y ^ { 2 } = 8 z$
C) $x ^ { 2 } + 8 y ^ { 2 } + z ^ { 2 } = 1$
D) $z ^ { 2 } - 8 y ^ { 2 } = x$
E) $z ^ { 2 } + x ^ { 2 } = - 8 y$

Question 10

Find parametric equations for the line through and

Question 11

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 14 x + 16 y - 8 z = 40$

A) $10.94$
B) $1.94$
C) $3.94$
D) $5.94$
E) $7.94$

Question 12

Find an equation for the surface obtained by rotating the line about the x-axis.

Question 13

Identify the planes that are perpendicular.

A) $10 x - y - z = 6 , - 9 x + y - 19 z = 2$
B) $x - 10 y = 6 , - 9 x - 19 z = 2$
C) $x + 10 y = 6 , - 9 x - y - 19 z = 2$
D) $x + 10 y = 6 , - 9 x - 19 z = 2$
E) $x + 10 y - z = 6 , - 9 x - y - 19 z = 2$

Question 14

Find the equation of the line through and perpendicular to the plane

Question 15

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ Round your answer to two decimal place.

A) $4.79$
B) $6.79$
C) $4.29$
D) $7.79$
E) $5.79$

Question 16

Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t .$

A) $x = 4 t + 2 , y = - 2 t + 4 , z = - 2 t + 5$
B) $x = 4 t + 2 , y = - 2 t + 4 , z = 0$
C) $x = 4 t , y = - 2 t , z = - 2 t + 5$
D) $x = 4 t , y = - 2 t + 4 , z = - 2 t$
E) $x = 4 t + 3 , y = - 2 t + 4 , z = - 2 t + 5$

Question 17

Find an equation of the plane that passes through the line of intersection of the planes and is perpendicular to the plane

Question 18

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

A) $\frac { 1 } { 6 }$
B) $\frac { 1 } { 3 }$ $\sqrt { 30 }$
C) $\frac { 5 \sqrt { 30 } } { 6 }$
D) $$\frac { 5 } { 6 }$$
E) $\sqrt { 30 }$

Question 19

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,5 x - 13 y + 10 z = 5$

A) $127 ^ { \circ }$
B) $12 ^ { \circ }$
C) $123 ^ { \circ }$
D) $7 ^ { \circ }$
E) none of these

Question 20

Find the point at which the line given by the parametric equations below intersects the plane. $$2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , \quad z = 7 t$$

A) $( 38 , - 10 , - 28 )$
B) $( - 43 , - 10,18 )$
C) $( - 38 , - 10,28 )$
D) $( 43 , - 10 , - 18 )$
E) $( 38 , - 10,28 )$

Question 21

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 7,5,1 \rangle , \mathbf { b } = ( - 5,2 , - 2 )$

A) $( - 8 , - 12 , - 19 \}$
B) $\{ - 12,39,9 \}$
C) $\{ - 12,9,39 \}$
D) $\{ 39 , - 12,9 \}$
E) $( - 7 , - 23,25 )$

Question 22

Find an equation of the plane that passes through the point and contains the line

Question 23

Find the values of x such that the vectors $\{ 2 x , x , 3 \}$ and $\{ 6 , x , 9 \}$ are orthogonal.

A) $x = 9 , x = - 9$
B) $x = 3 , x = 9$
C) $x = 3 , x = - 3$
D) $x = - 3 , x = - 9$
E) $x = - 3 , x = 9$

Question 24

Calculate the given quantities if

Question 25

Find parametric equations for the line through and parallel to the vector

Question 26

Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let ft, ft, , lb.

Question 27

Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N W at a speed of km/h. (This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the ground speed of the plane. Round the result to the nearest hundredth.

Question 28

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.

Question 29

A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is cm long.
Find the magnitude of the torque about P correct to two decimal places.
Let N, , .

Question 30

Let $v = 7 j$ and let u be a vector with length $5$ that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .

A) $| \mathbf { u } \times \mathbf { v } | = 5$
B) $| \mathbf { u } \times \mathbf { v } | = 7$
C) $| \mathbf { u } \times \mathbf { v } | = 35$
D) $| \mathbf { u } \times \mathbf { v } | = 157$
E) $| \mathbf { u } \times \mathbf { v } | = 1$

Question 31

Find an equation of the plane through the origin and parallel to the plane

Question 32

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

A) $64 \mathrm {~J}$
B) $74 \mathrm {~J}$
C) $84 \mathrm {~J}$
D) $94 \mathrm {~J}$
E) $104 \mathrm {~J}$

Question 33

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18$ , $| \mathbf { v } | = 3$ , $\angle \theta = 80 ^ { \circ }$ .

A) $69.179$ .
B) $69.179$
C) $53.179$ .
D) $75.179$
E) $73.179$

Question 34

A woman walks due west on the deck of a ship at mi/h. The ship is moving north at a speed of mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Question 35

Two forces $F _ { 1 } \text { and } F _ { 2 }$ with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth.

A) $$17.7 \mathrm { lb }$$
B) $13.7 \mathrm { lb }$
C) $15.7 \mathrm { lb }$
D) $19.7 \mathrm { lb }$
E) $11.7 \mathrm { lb }$

Question 36

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

A) $\mathbf { i } - 7 \mathbf { j } + 8 \mathbf { k }$
B) $- 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k }$
C) $\mathbf { i } - 8 \mathbf { j } + 7 \mathbf { k }$
D) $8 \mathbf { i } - 6 \mathbf { j } + 56 \mathbf { k }$
E) None of these

Question 37

Calculate the angle between a and b (correct to the nearest degree). $\mathbf { a } = 9 \mathbf { i } - 8 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

A) $98 ^ { \circ }$
B) $49 ^ { \circ }$
C) $39 ^ { \circ }$
D) $59 ^ { \circ }$
E) None of these

Question 38

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

A) $\frac { \pm ( 2 \mathbf { i } - 8 \mathbf { j } ) } { 65 }$
B) $\frac { \pm ( 2 \mathbf { i } + 8 \mathbf { j } ) } { 65 }$
C) $\frac { \pm ( i + 8 \mathbf { j } ) } { \sqrt { 65 } }$
D) $\frac { \pm ( \mathbf { i } + \mathbf { j } ) } { \sqrt { 65 } }$
E) $\frac { \pm ( \mathbf { i } - 8 \mathbf { j } ) } { 65 }$

Question 39

Find the point of intersection.

Question 40

The tension T at each end of the chain has magnitude N and makes an angle with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.

Question 41

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xz-plane and units to the right of it.

Question 42

Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.

Question 43

Draw a rectangular box with the origin and as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Question 44

Find the midpoint of the line segment joining the given points. (-5, 0, 2) and (-3, -2, 4)

A) ( $- 1$ , $$- 1$$ , $- 1$ )
B) ( $- 1$ , $1$ , $3$ )
C) ( $- 1$ , $1$ , $- 1$ )
D) ( $- 4$ , $- 1$ , $3$ )

Question 45

Find the length of the median of side AB of the triangle with vertices

Question 46

Find that the midpoint of the line segment from $P 1 ( - 12,12,9 )$ to $P 2 ( 2,8,3 )$ .

A) $( - 5 , - 10 , - 6 )$
B) $( - 5,10,6 )$
C) $( - 5 , - 10,6 )$
D) $( - 5,6,10 )$
E) $( - 5,10 , - 6 )$

Question 47

Find an equation of the sphere with center $( 6 , - 3,6 )$ that touches the xy-plane.

A) $( x + 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$
B) $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$
C) $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$
D) $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 6$
E) $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 40$

Question 48

Determine whether the given points are collinear. A (-3, -2, -3), B (-9, -5, 0), and C (-1, -1, -4)

A) Not collinear
B) Collinear

Question 49

Find the standard equation of the sphere with center C and radius r. C (3, -5, 3); r = 7

A) (x - 3)² + (y + 5)² + (z - 3)² = 7
B) (x - 3)² + (y + 5)² + (z - 3)² = 49
C) (x + 3)² + (y - 5)² + (z + 3)² = 49
D) (x + 3)² + (y - 5)² + (z + 3)² = 7

Question 50

Write inequalities to describe the solid rectangular box in the first octant bounded by the
planes , , and .

Question 51

a. Find an equation of the sphere that passes through the point and has center .
b. Find the curve in which this sphere intersects the xy-plane.

Question 52

Plot the given points in a three-dimensional coordinate system. (1, 2, 3)

A)
B)
C)
D)

Question 53

Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. A (-1, 0, 1), B (1, 1, -1), C (1, 1, 1)

A) 1, 1, $\sqrt { 2 }$ , both
B) 1, 2, $\sqrt { 5 }$ , neither
C) 3, 2, $\sqrt { 5 }$ , right
D) 1, 1, $$\sqrt { 2 }$$ , isosceles

Question 54

Find an equation of the sphere that passes through the point and has center .

Question 55

Suppose you start at the origin, move along the x-axis a distance of $6$ units in the positive direction, and then move downward a distance of $1$ units. What are the coordinates of your position?

A) $( 6,1,0 )$
B) $( 6,0 , - 1 )$
C) $( 6,0,1 )$
D) $( 6 , - 1,0 )$
E) $( 0,6,1 )$

Question 56

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 36 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

A) $C ( - 1 , - 12 , - 13 ) , r = 6$
B) $C ( - 1 , - 12 , - 13 ) , r = 6 \sqrt { 2 }$
C) $C ( - 1 , - 12 , - 13 ) , r = \sqrt { 2 }$
D) $C ( 1 , - 12 , - 13 ) , r = 6$
E) none of these

Question 57

Write an inequality to describe the region consisting of all points between (but not on) the
spheres of radius and centered at the origin.

Question 58

Find the distance from $( - 10,2 , - 8 )$ to the xy-planes.

A) 8
B) 16
C) 10
D) 2
E) 4

Question 59

Sketch the plane in a three-dimensional space represented by the equation. z = 2

A)
B)
C)
D)

Question 60

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 }$ + $y ^ { 2 }$ + $z ^ { 2 }$ - 6x + 4y = 0

A) ( -3, 2, 0), $\sqrt { 13 }$
B) ( 3, -2, 0), 13
C) ( -3, 2, 0), 13
D) ( 3, -2, 0), $\sqrt { 13 }$