Game Development Reference
In-Depth Information
This produces a vector orthogonal to both a and b . The magnitude of the cross
product is
sin θ,
where θ is the angle between a and b . The direction of the cross product is
determined by the right-hand rule: taking your right hand, point the first finger in
the direction of a and the middle finger along b . Your extended thumb will point
along the cross product.
Two useful identities to be aware of are the anticommutativity and bilinearity
of the cross product:
a
×
b
=
a
b
a ,
a × ( s b + t c )= s ( a × b )+ t ( a × c ) .
a
×
b =
b
×
1.2.5 Triple Product
There are two possible triple products for vectors. The first uses both the dot
product and the cross product and produces a scalar result. Hence it is known as
the scalar triple product :
c ) .
The scalar triple product measures the signed volume of the parallelepiped bounded
by the three vectors a , b ,and c . Thus, the following identity holds:
s = a
·
( b
×
a
·
( b
×
c )= b
·
( c
×
a )= c
·
( a
×
b ) .
The second triple product uses only the cross product and produces a vector result.
It is known as the vector triple product :
v = a
×
( b
×
c ) .
The vector triple product is useful for creating an orthogonal basis from linearly
independent vectors. One example basis is b , b
×
×
×
c ).
The following relationship between the vector triple product and dot product
is also helpful in derivations for rigid-body dynamics and geometric algorithms:
c ,and b
( b
a × ( b × c )=( a · c ) b ( a · b ) c .
1.2.6 Derivatives
We mentioned that vectors can act to represent rate of change. In particular, a
vector-valued function is the derivative of a point-valued function. If we take the
standard equation for a derivative of a function as in
x ( t + h )
x ( t )
x ( t ) = lim
h→ 0
,
h