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:

.

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 .

Read more about this topic:  Triple Product

Famous quotes containing the words triple and/or product:

    “And DANTE searched the triple spheres,
    Moulding nature at his will,
    So shaped, so colored, swift or still,
    And, sculptor-like, his large design
    Etched on Alp and Apennine.”
    —Ralph Waldo Emerson (1803–1882)

    “Poetry, whose material is language, is perhaps the most human and least worldly of the arts, the one in which the end product remains closest to the thought that inspired it.... Of all things of thought, poetry is the closest to thought, and a poem is less a thing than any other work of art ...”
    —Hannah Arendt (1906–1975)