Graphics Reference
In-Depth Information
u
×
v
v
=
b
.
h
= (
u
×
v
)
.
w
w
b
=
||
u
×
v
||
v
u
×
v
||
u
×
v
||
h
=
w
.
u
Figure 3.8
The scalar triple product (
u
×
v
) ·
w
is equivalent to the (signed) volume of the
parallelepiped formed by the three vectors
u
,
v
, and
w
.
The scalar triple product also remains constant under the cyclic permutation of its
three arguments:
(
u
×
v
)
·
w
=
(
v
×
w
)
·
u
=
(
w
×
u
)
·
v
.
Because of these identities, the special notation [
uvw
] is often used to denote a triple
product. It is defined as
[
uvw
]
=
(
u
×
v
)
·
w
=
u
·
(
v
×
w
).
This notation abstracts away the ordering of the dot and cross products and makes it
easy to remember the correct sign for triple product identities. If the vectors read
uvw
from left to right (starting at the
u
and allowing wraparound), the product identity is
positive, otherwise negative:
[
uvw
]
=
[
vwu
]
=
[
wuv
]
=−
[
uwv
]
=−
[
vuw
]
=−
[
wvu
] .
The scalar triple product can also be expressed as the determinant of the 3
×
3 matrix,
where the rows (or columns) are the components of the vectors:
u
1
−
u
2
+
u
3
=
u
1
u
2
u
3
v
1
v
2
v
3
w
1
w
2
w
3
v
2
v
3
w
2
w
3
v
1
v
3
w
1
w
3
v
1
v
2
w
1
w
2
[
uvw
]
=
=
u
·
(
v
×
w
).
=
Three vectors
u
,
v
, and
w
lie in the same plane if and only if [
uvw
]
0.
