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Figure 10:

Consider three vectors , , and . The scalar triple product is
defined
. Now,
is the vector area of
the parallelogram defined by and . So,
is the scalar area of this parallelogram times the component of in the direction
of its normal. It follows that
is
the volume of the parallelepiped defined by vectors , , and (see Fig. 10).
This volume is independent of how the triple product is formed from , ,
and , except that

(44) 
So, the ``volume'' is positive if , , and form a righthanded set
(i.e., if lies above the plane of and ,
in the sense determined from the righthand grip rule by rotating
onto ) and negative if they form a lefthanded set.
The triple product is unchanged if the dot and cross product operators are interchanged:

(45) 
The triple product is also invariant under any cyclic permutation of , ,
and ,

(46) 
but any anticyclic permutation causes it to change sign,

(47) 
The scalar triple product is zero if any
two of , , and are parallel, or if , , and
are coplanar.
If , , and are noncoplanar, then any vector can be
written in terms of them:

(48) 
Forming the dot product of this equation with
, we then obtain

(49) 
so

(50) 
Analogous expressions can be written for and . The parameters , ,
and are uniquely determined provided
:
i.e., provided that the three basis vectors are not coplanar.
Next: The vector triple product
Up: Vectors
Previous: Rotation
Richard Fitzpatrick
20060202