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 we fairies, that do run
    By the triple Hecate’s team
    From the presence of the sun,
    Following darkness like a dream,
    Now are frolic. Not a mouse
    Shall disturb this hallowed house.”
    —William Shakespeare (1564–1616)

    “In fast-moving, progress-conscious America, the consumer expects to be dizzied by progress. If he could completely understand advertising jargon he would be badly disappointed. The half-intelligibility which we expect, or even hope, to find in the latest product language personally reassures each of us that progress is being made: that the pace exceeds our ability to follow.”
    —Daniel J. Boorstin (b. 1914)