Graphics Reference
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A
A ￿ B
B
d A cos
B
d
FIGURE B.7
Computing the length of the projection of vector
A
onto vector
B
.
y
A
v ( v x ,v y ,v z )
u ( u x ,u y ,u z )
x
w ( w x , w y , w z )
z
A u
A ￿ u
A v
A ￿ v
A w
A ￿ w
A ( A x , A y , A z ) A u u A v v A w w
FIGURE B.8
Computing the coordinates of a vector in an auxiliary coordinate system.
B.2.4 Cross-product of two vectors
The cross-product ,or outer product , of two vectors can be defined using the determinant of a 3
3
matrix as shown in Equation B.24 , where i , j , and k are unit vectors in the directions of the principal
axes. Equation B.25 shows the definition as an explicit equation. The cross-product is not commutative
( Eq. B.26 ) , but it is associative ( Eq. B.27 ) .
i j k
A x A y A z
B x B y B z
A B ¼
(B.24)
i þ A z B x A x B z
k
A B ¼ A y B z A z B y
ð
Þj þ A x B y A y B x
(B.25)
A B ¼B A
ð
Þ¼ ðÞðÞ
(B.26)
 
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