James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

818 Chapter 12 Vectors and the Geometry of Space
The length of the cross product a 3 b is equal to the area of the parallelogram
determined by a and b.
ExamplE 3 Find a vector perpendicular to the plane that passes through the points
Ps1, 4, 6d, Qs22, 5, 21d, and Rs1, 21, 1d.
SOLUTION The vector PQ l 3 PR l is perpendicular to both PQ l and PR l and is therefore
perpendicular to the plane through P, Q, and R. We know from (12.2.1) that
PQ l − s22 2 1d i 1 s5 2 4d j 1 s21 2 6d k − 23i 1 j 2 7k
PR l − s1 2 1d i 1 s21 2 4d j 1 s1 2 6dk − 25 j 2 5k
We compute the cross product of these vectors:
Z
Z
PQ l 3 PR l i j k
− 23 1 27
0 25 25
− s25 2 35d i 2 s15 2 0d j 1 s15 2 0d k − 240 i 2 15 j 1 15k
So the vector k240, 215, 15 l is perpendicular to the given plane. Any nonzero scalar
multiple of this vector, such as k28, 23, 3 l, is also perpendicular to the plane. ■
ExamplE 4 Find the area of the triangle with vertices Ps1, 4, 6d, Qs22, 5, 21d,
and Rs1, 21, 1d.
SOLUTION In Example 3 we computed that PQ l 3 PR l − k240, 215, 15 l. The area of
the parallelogram with adjacent sides PQ and PR is the length of this cross product:
| PQ l 3 PR l | − ss240d 2 1 s215d 2 1 15 2 − 5s82
The area A of the triangle PQR is half the area of this parallelogram, that is, 5 2 s82 .
■
If we apply Theorems 8 and 9 to the standard basis vectors i, j, and k using − y2,
we obtain
i 3 j − k j 3 k − i k 3 i − j
j 3 i − 2k k 3 j − 2i i 3 k − 2j
Observe that
i 3 j ± j 3 i
Thus the cross product is not commutative. Also
whereas
i 3 si 3 jd − i 3 k − 2j
si 3 id 3 j − 0 3 j − 0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.