Graphics Reference
In-Depth Information
u
×
v
v
u
+
v
n
u
Figure 3.6
Given two vectors
u
and
v
in the plane, the cross product
w
(
w
=
u
×
v
)isa
vector perpendicular to both vectors, according to the right-hand rule. The magnitude of
w
is equal to the area of the parallelogram spanned by
u
and
v
(shaded in dark gray).
curved about the
w
vector such that the fingers go from
u
to
v
, the direction of
w
coincides with the direction of the extended thumb.
The magnitude of
u
×
v
equals the area of the parallelogram spanned by
u
and
v
,
with base
u
and height
v
sin
θ
(Figure 3.6). The magnitude is largest when the
vectors are perpendicular.
Those familiar with determinants might find it easier to remember the expression
for the cross product as the pseudo-determinant:
i
j k
u
2
u
3
u
1
u
3
u
1
u
2
u
×
v
=
u
1
u
2
u
3
=
i
−
j
+
k
,
v
2
v
3
v
1
v
3
v
1
v
2
v
1
v
2
v
3
where
i
(0, 0, 1) are unit vectors parallel to the
coordinate axes. The cross product can also be expressed in matrix form as the product
of a (skew-symmetric) matrix and a vector:
=
(1,0,0),
j
=
(0, 1, 0), and
k
=
⎡
⎤
⎡
⎤
0
−
u
3
u
2
v
1
v
2
v
3
⎣
⎦
⎣
⎦
.
u
×
v
=
u
3
0
−
u
1
−
u
2
u
1
0
It is interesting to note that the cross product can actually be computed using only
five multiplications, instead of the six multiplications indicated earlier, by express-
ing it as
u
×
v
=
(
v
2
(
t
1
−
u
3
)
−
t
4
,
u
3
v
1
−
t
3
,
t
4
−
u
2
(
v
1
−
t
2
)),
