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Math 2263 Multivariable Calculus
Quiz 1: 12.4-12.5
Name:
June 16, 2011
1. (a) Find two unit vectors orthogonal to both h0, 1, 2i and h1, −1, −2i.
Solution: The cross product of the vectors is orthogonal to both of them.
h0, 1, 2i × h1, −1, 2i = h0, 2, −1i
√
√
2 5
5
The unit vector in this direction is 0, 5 , − 5 . To get two unit vectors, we’ll
use this vector and the one in the opposite direction,
√
√ √ √ 2 5
5
2 5 5
0,
,−
,
and 0, −
.
5
5
5
5
(b) Use part of your work from (a) to find an equation for the line of intersection
of the planes y + 2z = 1 and x = 1 + y + 2z.
Solution: We find a point in the line by inspection. One possible point is
r0 = h2, 1, 0i. We find a direction vector by finding a vector orthogonal to both
normal vectors.
v = h0, 1, 2i × h1, −1, 2i = h0, 2, −1i
The line equation is
r = h2, 1, 0i + th0, 2, −1i.
2. Are the following lines parallel, skew, or intersecting?
Line 1: x = 4t, y = 2 − 4t, z = 6t + 5
Line 2: x = −6s, y = 6s + 3, z = 2 − 9s
Solution: Their directions vectors are scalar multiples, so they are parallel.