MATH 263C, Winter 2007, SOLUTION of HW 12.4
HW numbers 25, and 33 of 12.4 were already discussed in the class. Here are
solutions of number 15, number 29 and number 31 for you to study. You have to do
also problems 1-13 (odd), 23, 27. But these problems are not di±cult.
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15. Find two unit vectors that are orthogonal to vectors:
a = < 1; ¡1; 1 >; b = < 0; 4; 4 > :
ANSWER: We know that unit vectors are vectors that have length 1. We also know
that a £ b and b £ a are orthogonal to a and also to b. Hence the ¯rst unit vector is
a£b
b£a
u1 = ja£bj
: The second one is u2 = jb£aj
. Now you know how to ¯nd u1 and u2 . Do
¯nd these u1 ; u2 using the given a; b above.
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29. Find the volume of the parallelepiped determined by the vectors:
a = < 6; 3; ¡1 >; b = < 0; 1; 2 >; c = < 4; ¡2; 5 > :
ANSWER: We know that V = ja:(b £ c)j. Now
¯
¯
¯i j k¯
¯
¯
b £ c = ¯¯ 0 1 2 ¯¯ = 9i ¡ (¡8)j + (¡4)k = < 9; 8; ¡4 > :
¯ 4 ¡2 5 ¯
Hence V = < 6; 3; ¡1 > : < 9; 8; ¡4 >= j54 + 24 + 4j = 82:
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31. Find the volume of the parallelepiped with adjacent edges P Q; P R; P S where
P (2; 0; ¡1); Q(4; 1; 0); R(3; ¡1; 1); S(2; ¡2; 2) are 4 points in the 3-dimensional
space.
ANSWER. Let O be the origin of the (x; y; z)-coordinate system. Then the vectors
OP = < 2; 0; ¡1 >; OQ = < 4; 1; 0 >; OR = < 3; ¡1; 1 >; OS = < 2; ¡2; 2 > have the
initial point at O and the endpoints are P; Q; R; S: Hence the vectors P Q; P R; P S
are: P Q = OQ ¡ OP = < 2; 1; 1 >; P R = OR ¡ OP = < 1; ¡1; 2 >; P S = OS ¡ OP =
< 0; ¡2; 3 > :
Now the volume of the parallelepiped is:
V = jP Q:(P R¯ £ P S)j = ¯j < 2; 1; 1 > :(< 1; ¡1; 2 > £ < 0; ¡2; 3 >)j =
¯i j k¯
¯
¯
j < 2; 1; 1 > : ¯¯ 1 ¡1 2 ¯¯ j = j < 2; 1; 1 > : < 1; ¡3; ¡2 > j = j2 ¡ 3 ¡ 2j = 3:
¯ 0 ¡2 3 ¯
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