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8. a.
b. y 1 z 1 4 , 2
c. x 3t 3s 7, y t, z s,
s, tR
9. a. x 1 y 1 2 5 2 1
36 t,
12 t,
z t, tR
b. y 1 z t,
4 1
8 31
24 t, 12 t,
tR
10. a. These three planes meet at the point
11, 5, 32.
b. The planes do not intersect.
Geometrically, the planes form a
triangular prism.
c. The planes meet in a line through
the origin, with equation x t,
y 7t, z 5t, tR
11. 4.90
12. a. x
r ! 2y z 4 0
m ! 13,
n ! 1, 52 s12, 1, 02, sR
12, 1, 0211, 2, 12 0
Since the line’s direction vector is
perpendicular to the normal of the
plane and the point 13, 1,52 lies
on both the line and the plane, the
line is in the plane.
b.
c.
11, 1, 52
x 2y z 4 0
1 2112 152 4 0
The point 11, 1, 52 is on the
plane since it satisfies the equation
of the plane.
d. 7x 2y 11z 50 0
13. a.
b.
5.48
13, 0, 12
14. a.
b. r ! 12, 3, 02.
12, 3, 02 t11, 2, 12,
tR
15. a. 10x 9y 8z 16 0
b. about 0.45
16. a. 1
b. r ! 10, 0, 12 t14, 3, 72, tR
17. a. x 2, y 1, z 1
b. x 7 3t, y 3 t, z t, tR
18. a 2 3 , b 3 4 , c 1 2
19.
20.
a 4, 7 4 , 7 2 b
a 5 3 , 8 3 , 4 3 b
21. a. r ! a 45 4 , 0, 21 4 b
y 1 2 z t, tR
7 t,
t111, 2, 52, tR;
r ! a 37
2 , 0, 15 2 b
;
r ! t111, 2, 52, tR
17, 0, 12 t111, 2, 52,
tR;
z 1 5t, tR
b. All three lines of intersection found
in part a. have direction vector
111, 2, 52, and so they are all
parallel. Since no pair of normal
vectors for these three planes is
parallel, no pair of these planes is
coincident.
22. a 1 2 , 1, 1 3 b , a 1 2 , 1, 1 3 b , a 1 2 , 1, 1 3 b ,
a 1 2 , 1, 1 3 b , a 1 2 , 1, 1 3 b ,
a 1 a 1 and
2 , 1, 1 3 b 2 , 1, 1 3 b ,
a 1 2 , 1, 1 3 b
23. y 7 6 x2 3 2 x 2 3
24. a 29
7 , 4 7 , 33 7 b
25. A
26. a. r ! 5, B 2, C 4
11, 4, 62
t 15, 4, 32, tR
b. a 13
2 , 2, 3 2 b
c. about 33.26 units 2
27. 6x 8y 9z 115 0
Chapter 9 Test, p. 556
1. a. 13, 1, 52
b. 3 112 152 1 0
3 1 5 1 0
0 0
2. a.
13
or 1.08
12
b.
40
or 13.33
3
3. a.
b.
x 4t y 1 t z t, tR
5 , 5 ,
14, 0, 52
4. a. 11, 5, 42
b. The three planes intersect at the
point 11, 5, 42.
5. a. x 1 y 3t z t,
4 1 2 t 4 , 2 ,
tR
b. The three planes intersect at this line.
6. a. m 1, n 3
b. x 1, y 1 t, z t, tR
7. 10.20
Cumulative Review of Vectors,
pp. 557–560
1. a. about 111.0°
b. scalar projection: 14 ,
13
vector projection:
a 52
169 , 56
169 , 168 169 b
c. scalar projection: 14 ,
3
vector projection:
a 28
9 , 14 9 , 28 9 b
2. a. x 8 4t, y t, z 3 3t,
tR
b. about 51.9°
1
3. a.
2
b. 3
3
c.
2
4. a. 7i ! 19j ! 14k !
b. 18
5. x-axis: about 42.0°, y-axis: about
111.8°, z-axis: about 123.9°
6. a. 17, 5, 12
b. 142, 30, 62
c. about 8.66 square units
d. 0
7. a 1 and a 1 2 , 1
2 , 1 2 , 0b
2 , 0b
8. a. vector equation: Answers may vary.
r ! 12, 3, 12 t11, 5, 22, tR;
parametric equation:
x 2 t, y 3 5t,
z 1 2t, tR
b. If the x-coordinate of a point on the
line is 4, then 2 t 4, or
t 2. At t 2, the point on
the line is 12, 3, 12 211, 5, 22
14, 13, 32. Hence,
C14, 13, 32 is a point on the line.
9. The direction vector of the first line is
11, 5, 22 and of the second line is
11, 5, 22 11, 5, 22. So they
are collinear and hence parallel.
The lines coincide if and only if for
any point on the first line and second
line, the vector connecting the two
points is a multiple of the direction
vector for the lines. (2, 0, 9) is a point
on the first line and 13, 5, 102 is a
point on the second line.
12, 0, 92 13, 5, 102 11, 5, 12
k11, 5, 22 for kR. Hence, the
lines are parallel and distinct.
x 5 7 t, Answers 687
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