Graphics Reference
In-Depth Information
2.10.2 The vector triple product
One of the vector triple products in Table 2.3 is a
×
b
×
c , and there are three questions we
can ask about it:
1. Is a
×
b
×
c equivalent to an expression involving dot products?
2. Does a
×
b
×
c have any connection with a
×
b
×
c ?
3. Does a
×
b
×
c have any geometric meaning?
a
b × c
b × c
b
b
c
c
a
×
(b
×
c)
(b)
(a)
Figure 2.27.
Figure 2.27(a) shows part of the triple product, where we see vectors b and c sharing a common
plane and the cross product b
c perpendicular to this plane. The third vector a can either
reside in this plane or intersect it. First, let's consider the scenario where it intersects the plane,
as shown in Fig. 2.27(b).
If we take the cross product of a with b
×
×
c , the result a
×
b
×
c must be perpendicular to a
and b
c , which means that it must be in the original plane containing b and c . If you find this
difficult to visualise, don't worry. Try this: if a vector exists perpendicular to some plane, and
we have to compute a vector product with it and some other vector, the result of this product
must be perpendicular to the original vector, which must be in the perpendicular plane! Thus,
we see in Fig. 2.27(b) the result of a
×
c lying in the plane containing b and c .
Now let's consider the scenario where a resides in the plane containing b and c . Well, if we
think about it, it is no different from the previous scenario, because the result a
×
b
×
×
b
×
c must
be perpendicular to b
×
c , which is the plane! The same is true for a
×
b
×
c .
It can be shown that
a
×
b
×
c
=
a
·
c b
a
·
b c
(2.20)
and
a
×
b
×
c
=
a
·
c b
b
·
c a
(2.21)
 
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