Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 418

y

A B

B

A

x

z

FIGURE B.9
Vector formed by the cross product of two vectors.

The direction of A B is perpendicular to both A and B ( Figure B.9 ) , and the direction is determined
by the right-hand rule (if A and B are in right-hand space). If the thumb of the right hand is put in the
direction of the first vector ( A ) and the index finger is put in the direction of the second vector ( B ), the
cross-product of the two vectors will be in the direction of the middle finger when it is held perpen-
dicular to the first two fingers.
The magnitude of the cross-product is the length of one vector times the length of the other vector
times the sine of the angle between them ( Eq. B.28 ) . A zero vector will result if the two vectors are
colinear or if either vector is a zero vector ( Eq. B.29 ) . This relationship is useful for determining
the sine of the angle between two vectors ( Figure B.10 ) and for computing the perpendicular distance
from a point to a line ( Figure B.11 ).

jA Bj¼jAjjBj sin y (B.28)

A

A B

sin

A B

B

FIGURE B.10
Using the cross-product to compute the sine of the angle between two vectors.

P

s P P 1 sin

( P P 1 ) ( P 2

P 1 )

s

P 2 P 1

P 2

P 1

FIGURE B.11