Graphics Reference
In-Depth Information
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
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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