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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.12.) This volume is independent of how the triple product is formed from , , and , except that

 (A.53)

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 a right-hand circulation 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,

 (A.54)

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

 (A.55)

but any anti-cyclic permutation causes it to change sign,

 (A.56)

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

If , , and are non-coplanar then any vector can be written in terms of them: that is,

 (A.57)

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

 (A.58)

so

 (A.59)

Analogous expressions can be written for and . The parameters , , and are uniquely determined provided : that is, provided the three vectors are non-coplanar.

Next: Vector Triple Product Up: Vectors and Vector Fields Previous: Rotation
Richard Fitzpatrick 2016-03-31