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

Section 12.4 The Cross Product 817
a rotation (through an angle less than 180°) from a to b, then your thumb points in the
direction of a 3 b.
Now that we know the direction of the vector a 3 b, the remaining thing we need to
complete its geometric description is its length | a 3 b | . This is given by the following
theorem.
9 Theorem If is the angle between a and b (so 0 < < ), then
| a 3 b | − | a | | b | sin
TEC Visual 12.4 shows how a 3 b
changes as b changes.
Proof From the definitions of the cross product and length of a vector, we have
| a 3 b | 2 − sa 2 b 3 2 a 3 b 2 d 2 1 sa 3 b 1 2 a 1 b 3 d 2 1 sa 1 b 2 2 a 2 b 1 d 2
− a 2 2 b 2 3 2 2a 2 a 3 b 2 b 3 1 a 2 3 b 2 2 1 a 2 3 b 2 1 2 2a 1 a 3 b 1 b 3 1 a 2 1 b 2 3
1 a 2 1 b 2 2 2 2a 1 a 2 b 1 b 2 1 a 2 2 b 2 1
− sa 2 1 1 a 2 2 1 a 2 3dsb 2 1 1 b 2 2 1 b 2 3 d 2 sa 1 b 1 1 a 2 b 2 1 a 3 b 3 d 2
− | a | 2 | b | 2 2 sa ? bd 2
− | a | 2 | b | 2 2 | a | 2 | b | 2 cos 2 (by Theorem 12.3.3)
− | a | 2 | b | 2 s1 2 cos 2 d
− | a | 2 | b | 2 sin 2
Taking square roots and observing that ssin 2 − sin because sin > 0 when
0 < < , we have
| a 3 b | − | a | | b | sin ■
Geometric characterization of a 3 b
Since a vector is completely determined by its magnitude and direction, we can now
say that a 3 b is the vector that is perpendicular to both a and b, whose orientation is
determined by the right-hand rule, and whose length is | a | | b | sin . In fact, that is
exactly how physicists define a 3 b.
10 Corollary Two nonzero vectors a and b are parallel if and only if
a 3 b − 0
Proof Two nonzero vectors a and b are parallel if and only if − 0 or . In either
case sin − 0, so | a 3 b | − 0 and therefore a 3 b − 0.
■
b b sin ¨
¨
a
FIGURE 2
The geometric interpretation of Theorem 9 can be seen by looking at Figure 2. If a
and b are represented by directed line segments with the same initial point, then they
determine a parallelogram with base | a | , altitude | b | sin , and area
A − | a | (| b | sin ) − | a 3 b |
Thus we have the following way of interpreting the magnitude of a cross product.
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