CHAPTER 10 REVIEW ■ 589
11. (a) Find a vector perpendicular to the plane through the
points , , and .
(b) Find the area of triangle .
12. A constant force moves an object along the line segment from to . Find the work done if the distance is measured in meters and the force in newtons.
13. A boat is pulled onto shore using two ropes, as shown in the diagram. If a force of 255 N is needed, find the magnitude of the force in each rope.
14. Find the magnitude of the torque about if a 50-N force is applied as shown.
15–17 ■ Find parametric equations for the line.
15. The line through and
16. The line through and parallel to the line
17. The line through and perpendicular to the plane
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
18 –20 ■ Find an equation of the plane.
18. The plane through and parallel to 19. The plane through , , and
20. The plane through that contains the line , ,
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
21. Find the point in which the line with parametric equations , , intersects the plane
.
22. Find the distance from the origin to the line , , .z 1 2t
y 2 t
x 1 t 2 xx 2 t y z 2y 1 3t z 4t
z 1 3t y 3 t
x 2t
1, 2, 2
6, 3, 1
4, 0, 2
3, 1, 1
x 4y 3z 1
2, 1, 0
2x y 5z 122, 2, 4
1
3x 4 12y z 21, 0, 1
1, 1, 5
4, 1, 2
P 40 cm
50 N 30°
P 20°
30°
255 N
5, 3, 8
1, 0, 2
F 3i 5j 10k ABC
C1, 4, 3
B2, 0, 1
A1, 0, 0
1. (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 -plane.
(c) Find the center and radius of the sphere
2. Copy the vectors in the figure and use them to draw each of the following vectors.
(a) (b) (c) (d)
3. If u and v are the vectors shown in the figure, find and . Is u v directed into the page or out of it?
4. Calculate the given quantity if
(a) (b)
(c) (d)
(e) ( f )
(g) (h)
(i) ( j)
(k) The angle between and (correct to the nearest degree)
5. Find the values of such that the vectors and are orthogonal.
6. Find two unit vectors that are orthogonal to both and .
7. Suppose that . Find
(a) (b)
(c) (d)
8. Show that if , , and are in , then
9. Find the acute angle between two diagonals of a cube.
10. Given the points , , , and
, find the volume of the parallelepiped with adja- cent edges AB, AC, and AD.
D0, 3, 2
C1, 1, 4
B2, 3, 0
A1, 0, 1
2
V3
c b a
u v v v u w
u w v
u v w
u v w 2 i 2j 3k
j 2k
2x, 4, x x 3, 2, x
b a
projab compab
a c c
a b c
b c a ba b
b2 a 3b
c j 5k b 3i 2j k
a i j 2k
45°
| v |=3
| u |=2
u v u va
b
2 a b
12a a b
a b
x2 y2 z2 8x 2y 6z 1 0 yz
1, 2, 1
6, 2, 3
E X E R C I S E S
41. If , evaluate . 42. Let be the curve with equations , ,
. Find (a) the point where intersects the -plane, (b) parametric equations of the tangent line at , and (c) an equation of the normal plane to at .
43. Use Simpson’s Rule with to estimate the length of the arc of the curve with equations , , ,
.
44. Find the length of the curve ,
.
45. The helix intersects the curve
at the point . Find the angle of intersection of these curves.
46. Reparametrize the curve
with respect to arc length measured from the point in the direction of increasing .
47. For the curve given by , find (a) the unit tangent vector, (b) the unit normal vector, and (c) the curvature.
48. Find the curvature of the ellipse , at the points and .
49. Find the curvature of the curve at the point .
;50. Find an equation of the osculating circle of the curve at the origin. Graph both the curve and its osculating circle.
51. A particle moves with position function
. Find the velocity, speed, and acceleration of the particle.
52. A particle starts at the origin with initial velocity
. Its acceleration is .
Find its position function.
53. An athlete throws a shot at an angle of to the horizontal at an initial speed of 43 fts. It leaves his hand 7 ft above the ground.
(a) Where is the shot 2 seconds later?
(b) How high does the shot go?
(c) Where does the shot land?
54. Find the tangential and normal components of the accelera- tion vector of a particle with position function
55. Find the curvature of the curve with parametric equations
y
y
0t cos(
122)
d xy
0t sin(
122)
drt t i 2t j t2k 45
at 6t i 12t2j 6t k i j 3k
rt t ln t i t j etk y x4 x2
1, 1
y x4
0, 4
3, 0
y 4 sin t x 3 cos t
rt
13t3,12t2, tt
1, 0, 1
rt eti etsin t j etcos t k
1, 0, 0
r2t 1 ti t2j t3k
r1t cos t i sin t j t k 0 t 1
rt 2t32, cos 2t, sin 2t
0 t 3 x t2 y t3 z t4
n 6
1, 1, 0
C
1, 1, 0
xz z ln t C
y 2t 1 x 2 t3
C
x
01rt dt rt t2i t cost j sin t k23. Determine whether the lines given by the symmetric equations
and
are parallel, skew, or intersecting.
24. (a) Show that the planes and
are neither parallel nor perpendicular.
(b) Find, correct to the nearest degree, the angle between these planes.
25. Find the distance between the planes and .
26 –34 ■ Identify and sketch the graph of each surface.
26. 27.
28. 29.
30.
31.
32.
33.
34.
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
35. An ellipsoid is created by rotating the ellipse
about the -axis. Find an equation of the ellipsoid.
36. A surface consists of all points such that the distance from to the plane is twice the distance from to the point . Find an equation for this surface and identify it.
37. (a) Sketch the curve with vector function
(b) Find and .
38. Let .
(a) Find the domain of . (b) Find . (c) Find .
39. Find a vector function that represents the curve of intersec- tion of the cylinder and the plane .
;40. Find parametric equations for the tangent line to the curve
, , at the point .
Graph the curve and the tangent line on a common screen.
(
1,s3, 2)
z 2 sin 3t y 2 sin 2t
x 2 sin t
x z 5 x2 y2 16
rt
limtl 0rt
r
rt s2 t , et 1t, lnt 1
rt
rt
t 0 rt t i cost j sin t k
0, 1, 0
P y 1
P
P x 4x2 y2 16
x y2 z2 2y 4z 5 4x2 4y2 8y z2 0 y2 z2 1 x2
4x2 y2 4z2 4 4x y 2z 4
x2 y2 4z2 y z2
x z x 3
3x y 4z 24
3x y 4z 2 2x 3y 4z 5
x y z 1 x 1
6 y 3
1 z 5 2 x 1
2 y 2
3 z 3 4
590 ■ CHAPTER 10 VECTORS AND THE GEOMETRY OF SPACE
