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Scalar Triple Product
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 multiplied by the component of in the direction
of its normal. It follows that
is
the volume of the parallelepiped defined by vectors , , and see Figure A.106.
This volume is independent of how the triple product is formed from , ,
and , except that

(1310) 
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 a righthand circulation 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,

(1311) 
The triple product is also invariant under any cyclic permutation of , ,
and ,

(1312) 
but any anticyclic permutation causes it to change sign,

(1313) 
The scalar triple product is zero if any
two of , , and are parallel, or if , , and
are coplanar.
Figure A.106:
A vector parallelepiped.

If , , and are noncoplanar then any vector can be
written in terms of them: i.e.,

(1314) 
Forming the dot product of this equation with
, we then obtain

(1315) 
so

(1316) 
Analogous expressions can be written for and . The parameters , ,
and are uniquely determined provided
:
i.e., provided that the three vectors are noncoplanar.
Next: Vector Triple Product
Up: Vector Algebra and Vector
Previous: Rotation
Richard Fitzpatrick
20110331