Graphics Reference
In-Depth Information
3.3.6
Algebraic Identities Involving Cross Products
Given scalars
r
and
s
and vectors
u
,
v
,
w
, and
x
, the following cross product identities
hold:
u
×
v
=−
(
v
×
u
)
u
×
u
=
0
u
×
0
=
0
×
u
=
0
·
×
=
·
×
=
×
u
(
u
v
)
v
(
u
v
)
0 (that is,
u
v
is perpendicular to both
u
and
v
)
u
·
(
v
×
w
)
=
(
u
×
v
)
·
w
u
×
(
v
±
w
)
=
u
×
v
±
u
×
w
(
u
±
v
)
×
w
=
u
×
w
±
v
×
w
(
u
×
v
)
×
w
=
w
×
(
v
×
u
)
=
(
u
·
w
)
v
−
(
v
·
w
)
u
(a vector in the plane of
u
and
v
)
u
×
(
v
×
w
)
=
(
w
×
v
)
×
u
=
(
u
·
w
)
v
−
(
u
·
v
)
w
(a vector in the plane of
v
and
w
)
u
×
v
=
u
v
sin
θ
(
u
×
v
)
·
(
w
×
x
)
=
(
u
·
w
)(
v
·
x
)
−
(
v
·
w
)(
u
·
x
) (Lagrange's identity)
r
u
×
s
v
=
rs
(
u
×
v
)
u
×
(
v
×
w
)
+
v
×
(
w
×
u
)
+
w
×
(
u
×
v
)
=
0
(Jacobi's identity)
The Lagrange identity is particularly useful for reducing the number of operations
required for various geometric tests. Several examples of such reductions are found
in Chapter 5.
3.3.7
The Scalar Triple Product
The expression (
u
w
occurs frequently enough that it has been given a name
of its own:
scalar triple product
(also referred to as the
triple scalar product
or
box
product
). Geometrically, the value of the scalar triple product corresponds to the
(signed) volume of a parallelepiped formed by the three independent vectors
u
,
v
,
and
w
. Equivalently, it is six times the volume of the tetrahedron spanned by
u
,
v
,
and
w
. The relationship between the scalar triple product and the parallelepiped is
illustrated in Figure 3.8.
The cross and dot product can be interchanged in the triple product without
affecting the result:
×
v
)
·
(
u
×
v
)
·
w
=
u
·
(
v
×
w
).
