Chapter 13
1.
Find the length of the median of side AB of the triangle with vertices A(2, 8,  7), B(6,  8,  3)
and C (2,  4, 9) .
2.
Find the center and radius of the sphere.
x 2  2 x  38  y 2  8 y  z 2  6 z  0
3.
Write an equation or inequality that represents a plane containing all points with equal x and z
coordinates.
4.
Write an equation or inequality that represents all points on or inside a circular cylinder of radius 1
with the y-axis as its axis.
5.
Write inequalities to describe the solid rectangular box in the first octant bounded by the planes
x = 9 , y = 6 , and z = 7 .
6.
Find the sum of the given vectors.
3, 8,  5 , 2, 9, 3
7.
Find the values of x such that the vectors
4 x, x, 3 and
8.
A bicycle pedal is pushed by a foot with a force of aN as shown. The shaft of the pedal is 15 cm long.
2, x, 5
are orthogonal.
Find the magnitude of the torque about P correct to two decimal places. Let
7 .
9.
a = 70,
b = 58  ,
Write the equation in cylindrical coordinates.
z  3x 2  3 y 2
10. Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as
shown. Let a = 1 ft, b = 1 ft, c = 30  , d = 44 lb.
c=
11. Find parametric equations for the line through the points (5, 1, 10) and (6,  9, 6) .
12. Find an equation of the plane that passes through the point (4, 0, 2) and contains the line
x  10  3t, y  10  8t, z  6  7t .
13. Find an equation of the plane that passes through the line of intersection of the planes x  z  2 and
y  3z  7 , and is perpendicular to the plane 5x  3 y  2z  8 .
14. Find an equation of the plane with x-intercept = 1, y-intercept = 10, and z-intercept = -10.
15. Find parametric equations for the line through the point (4, 5, 2) that is parallel to the plane
x  y  z  15 and perpendicular to the line x  15  t, y  12  t, z  3t .
Select the correct answer.
a.
x  4t  4, y  2t  5, z  2t  2
b.
x  2t  4, y  4t  5, z  2t  2
c.
x  4t  4, y  2t  5, z  2t  2
d.
x  4t, y  2t  5, z  2t
e.
x  4t  4, y  2t, z  2t  5
16. Find the distance between the planes.
5x  2 y  z  1  0 , 5 x  2 y  z  4  0
Select the correct answer.
30
5
a.
b.
5 6
c.
5
6
d.
30
6
e.
1
6
2
17. Find an equation for the surface obtained by rotating the parabola y  x about the y-axis.
Select the correct answer.
y2  x2  z
a.
b.
z 2  x2  y
c.
z 2  x2  y
d.
z 2  x2  y 2  1
e.
z  x  y 1
2
2
2
2
2
18. An ellipsoid is created by rotating the ellipse 4 x  y  16 about the x-axis. Find the equation of the
ellipsoid.
19. Change from rectangular to cylindrical coordinates.
( 4, - 4, 2 )
20. Write the equation in spherical coordinates.
x2  y 2  2 y
1.
248
2.
C 1,  4, 3 , r  8
3.
xz
4.
x2  z 2  1
5.
0  x  9, 0  y  6, 0  z  7
6.
 1, 17 , 2
7.
x  3, x  5
8.
9.52
9.
z  3r 2 cos(2 )
10. 60.11
11. x  5  11t , y  1  10t , z  10  4t
12. 6x  66 y  78z  132
13. 3x  7 y  18z  55
14. 100x  10 y  10z  100
15. a
16. d
17. b
2
2
2
18. 4 x  y  z  16
7


, 2
19.  4 2,
4


20.  sin   2 sin
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 8 units. What are the coordinates of your position?
Select the correct answer.
a.
b.
c.
(0, 4, 8)
d.
(4, 8, 0)
(4, 0, 8)
(4, 8, 0)
e.
2.
(4, 0, 8)
Find the length of the side AB of the triangle with vertices A(5, 5, 10) , B(3, 6, 1) and
C(9, 2, 1) .
3.
Find an equation of the sphere that passes through the point (7, 4, 9) and has center (4, 5, 5) .
4.
Given the following equation, find the radius of the sphere.
x 2  y 2  z 2  6x  2 y  4z
5.
Find the midpoint of the line segment from P1(8, 10,  3) to P2(2, 10, 7) .
Select the correct answer.
a.
e.
6.
(5, 0, 5)
(3, 10, 2)
b.
(3,  10,  2)
c.
(5, 0,  5)
d.
(3, 10, 2)
Write an equation or inequality that represents a plane containing all points with equal x and y
coordinates.
Select the correct answer.
a.
7.
8.
yz
b.
x y
c.
x y
d.
x y
Write inequalities to describe the solid rectangular box in the first octant bounded by the planes
x = 4, y = 8, and z = 5.
Find 9a + 4b.
a = 2i - 9j + 9k , b = 5i + 2j + 10k
9.
e.
Find a  b , if a  5i  13j  8k and b  14i  3k .
10. Find the angle between the vectors, if
a  12, 0
and b  12, 12 .
11. Find the cosine of the angle between the following vectors.
a = 2j + 6k
b = 6i + 4j + 3k
12. Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.
P ( 1, 2, 3 ),
Q ( 3, 5, 4 ),
R ( 3, 2, 5 ),
S ( 4, 2, 3 )
yz
13. Find a parametric equation for the line through the point (6, 9, 3) and parallel to the
vector 7, 3, 7 .
14. Find the point of intersection.
x  17 y  58 z  23


3
8
2
x  49 y  26
L2 :

 z  15
7
4
L1 :
15. Find an equation of the plane that passes through the line of intersection of the planes x  z  10 and
y  5z  9 , and is perpendicular to the plane 3x  2 y  2z  7 .
16. Find an equation of the plane with x-intercept = 9, y-intercept = 1, and z-intercept = -3.
17. Find an equation for the surface obtained by rotating the line x = 6y about the x-axis.
18. Change from rectangular to cylindrical coordinates.
( 9, - 9, 1 )
19. Identify the correct statement(s) below.
Select all that apply.
a.
b.
r  3 is a circular cylinder, radius 3, axis the z-axis
  7 is a circular cylinder, radius 7, axis the z-axis
c.

d.

e.

9

is a half-cone
is a half-cone
5
none of these
20. The sketch of the solid is given below. Given a = 2, write the inequalities that describe it.
Select the correct answer.
a.
b.
c.
d.
e.
r2  z  2  r2
r2  z  2  r2
r2  z  2
r  z  2r
r2  2  z  r2
1.
a
2.
3.
306
( x  4)2  ( y  5)2  ( z  5)2  398
4.
14
5.
d
6.
c
7.
0  x  4, 0  y  8, 0  z  5
8.
38i – 73j + 121k
9.
46
10.

4
11. 0.53
12. 18
13.
x  6  7t, y  9  3t, z  3  7t
14.
(7, 6, 7)
15.
8x  5 y  17z  125
16.
3x  27 y  9z  27
17.
z2  y2 
18.
7 

, 1
9 2,
4


19. a, c
20. b
x2
36