Lagrange's formula

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In mathematics, Lagrange's formula is a formula relating the cross product to the dot product of vectors in R3. It states,

a × (b × c) = b(a · c) − c(a · b),

which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. A special case regarding gradients and useful in vector calculus, is

\begin{matrix} \nabla \times (\nabla \times \mathbf{f}) &=& \nabla (\nabla \cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\ &=& \mbox{grad }(\mbox{div } \mathbf{f} ) - \mbox{laplacian } \mathbf{f}. \end{matrix}

This is a special case of the more general Laplace-de Rham operator Δ = dδ + δd.

Another useful identity of Lagrange is

|a \times b|^2 + |a \cdot b|^2 = |a|^2 |b|^2.

This is a special case of the multiplicativity | vw | = | v | | w | of the norm in the quaternion algebra.

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