familiar mnemonic "BAC − CAB" is obtained, as in “back of the cab”. Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as: From Lagrange's formula it follows that the vector triple product satisfies: which is the Jacobi identity for the cross product. Another useful formula follows: 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: This can be also regarded as a special case of the more general Laplace–de Rham operator . Proof The component of is given by: Similarly, the and components of are given by: By combining these three components we obtain: Using geometric algebra If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product b∧c, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction can be used, so the formula becomes The proof follows from the properties of the contraction. The result is the same vector as calculated using a × (b × c). Interpretations Tensor calculus In tensor notation, the triple product is expressed using the Levi-Civita symbol: and referring to the -th component of the resulting vector. This can be simplified by performing a contraction on the Levi-Civita symbols, where is the Kronecker delta function ( when and when ) and is the generalized Kronecker delta function. We can reason out this identity by recognizing that the index will be summed out leaving only and . In the first term, we fix and thus . Likewise, in the second term, we fix and thus . Returning to the triple cross product, Vector calculus Consider the flux integral of the vector field across the parametrically-defined surface : . The unit normal vector to the surface is given by , so the integrand is a scalar triple product. See also Quadruple product Vector algebra relations Notes References External links Khan Academy video of the proof of the triple product expansion Articles containing proofs Mathematical identities Multilinear algebra Operations on vectors Ternary operations