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## The 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 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 right-handed set (i.e., if lies above the plane of and , in the sense determined from the right-hand grip rule by rotating onto ) and negative if they form a left-handed 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 anti-cyclic permutation causes it to change sign,
 (47)

The scalar triple product is zero if any two of , , and are parallel, or if , , and are co-planar.

If , , and are non-coplanar, 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 co-planar.

Next: The vector triple product Up: Vectors Previous: Rotation
Richard Fitzpatrick 2006-02-02