33.40
For nonzero vectors A and B, find a necessary and sufficient condition that A • B = |A| |B|.
A • B = |A| |B| cos 0, where 0 is the angle between A and B. Hence, A • B = |A| |B| if and only if
cos 0 = 1, that is, if and only if 6 = 0, which is equivalent to A and B having the same direction. (In other
words, A = «B for some positive scalar u.)
Fig. 33-11
33.41
Derive the law of cosines by vector methods.
In the triangle of Fig. 33-11, let |A| = a, \B\ = b, |C| = c. Then a
2 = A- A = (B -C) -(B -C) =
B • B + C • C - 2B • C = b
1 4- c
2 - 2bc cos 0.
PLANAR VECTORS
273
For nonzero vectors A and B, find a necessary and sufficient condition that A • B = |A| |B|.
A • B = |A| |B| cos 0, where 0 is the angle between A and B. Hence, A • B = |A| |B| if and only if
cos 0 = 1, that is, if and only if 6 = 0, which is equivalent to A and B having the same direction. (In other
words, A = «B for some positive scalar u.)
Fig. 33-11
33.41
Derive the law of cosines by vector methods.
In the triangle of Fig. 33-11, let |A| = a, \B\ = b, |C| = c. Then a
2 = A- A = (B -C) -(B -C) =
B • B + C • C - 2B • C = b
1 4- c
2 - 2bc cos 0.
PLANAR VECTORS
273