Triple Product - Vector Triple Product

The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationships hold:

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The first formula is known as triple product expansion, or Lagrange's formula, although the latter name is ambiguous (see disambiguation page). Its right hand member is easier to remember by using the mnemonic “BAC minus CAB”, provided one keeps in mind which vectors are dotted together. A proof is provided below.

These formulas are very useful in simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product identity:

\begin{align} \nabla \times (\nabla \times \mathbf{f}) & {}= \nabla (\nabla \cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\ & {}= \mbox{grad }(\mbox{div } \mathbf{f} ) - \nabla^2 \mathbf{f}. \end{align}

This can be also regarded as a special case of the more general Laplace-de Rham operator .

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