Thomas Calculus 13th [Solutions]

L1 & L3:
x 3 2t 3 2r 2t 2r 0 t r 0
y 1 4t 2 r 4t r 3 4t r 3
Section 12.5 Lines and Planes in Space 905
3t
3 t 1
and r 1 on L1, z 2
while on L3, z 0 L1 and L2 do not intersect. The direction of L1 is 1 2 i 4 j k while the direction of
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L3 is 1 3 2 i j 2 k and neither is a multiple of the other; hence L1 and L3 are skew.
x 1 2t 2 s 2t s 1
62. L1 & L2: 5s
3 s 3
y 1 t 3s t 3s
1
5
and t 4 on L1, z 12 while on L2,
5
5
z 1 3 2 L1 and L2 do not intersect. The direction of L1 is 1
5 5
2 i j 3 k while the direction of L2 is
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1
11
L2 & L3:
i 3j k and neither is a multiple of the other; hence, L1 and L2 are skew.
x 2 s 5 2r s 2r
3
y 3s 1 r 3s r 1
z 2 L2 and L3 intersect at (1, 3, 2).
5s
5 s 1
and r 2 on L2, z 2 and on L3,
L1 & L3: L1 and L3 have the same direction 1 2 i j 3 k ; hence L1 and L3 are parallel.
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63. x 2 2 t, y 4 t, z 7 3 t; x 2 t, y 2 1 t, z 1 3 t
2 2
64. 1( x 4) 2( y 1) 1( z 5) 0 x 4 2y 2 z 5 0 x 2y z 7;
2( x 3) 2 2( y 2) 2( z 0) 0 2x 2 2y 2z
7 2
65. x 0 t 1 , y 1 , z 3 0, 1 , 3 ; y 0 t 1, x 1, z 3 ( 1, 0, 3);
2 2 2 2 2
z 0 t 0, x 1, y 1 (1, 1, 0)
66. The line contains (0, 0, 3) and 3, 1, 3 because the projection of the line onto the xy -plane contains the
origin and intersects the positive x -axis at a 30° angle. The direction of the line is 3i j 0k the line in
question is x 3 t, y t, z 3.
67. With substitution of the line into the plane we have 2(1 2 t) (2 5 t) ( 3 t) 8 2 4t 2 5t 3t
8
4t 4 8 t 1 the point ( 1, 7, 3) is contained in both the line and plane, so they are not parallel.
68. The planes are parallel when either vector A1 i B1 j C1k or A2i B2 j C2k is a multiple of the other or when
A1 i B1 j C1k A2i B2 j C2 k 0 . The planes are perpendicular when their normals are perpendicular,
or A1 i B1 j C1k A2i B2 j C2k
0.
69. There are many possible answers. One is found as follows: eliminate t to get t x 1 2 y z 3
2
x 1 2 y and 2 y z 3 x y 3 and 2y z 7 are two such planes.
2
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