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Triple Products of Vectors

There are two triple products of vectors. The first is the scalar triple product:

$\displaystyle \vA \cdot (\vB \times \vC) $

If $ \vA$ , $ \vB$ and $ \vC$ are all length vectors, this represents the volume of parallelopiped formed by the vectors.

The second is the vector triple product:

$\displaystyle \vA \times (\vB \times \vC) = \vB(\vA\cdot\vC) - \vC(\vA\cdot\vB) $

This last identity is called the BAC-CAB (palindromic) rule. It is tedious but straightforward to prove it for Cartesian vector components. First, however, we would like to introduce two special tensor forms that greatly simplify the algebra of both dot and cross products and enable us to prove various vector identities using algebra instead of a tedious enumeration of terms.

Robert G. Brown 2017-07-11