(a)Find a vector perpendicular to the plane through thepoints , , and .(b)Find the area of triangle .

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





a b





2 a 3b

c j  5k b 3i  2j  k

a i  j  2k

45°

| v |=3

| u |=2





a

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