Editing Cross product (section)

=== Triple product expansion ===
{{Main|Triple product}}

The cross product is used in both forms of the triple product. The [[scalar triple product]] of three vectors is defined as

:<math>\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}), </math>

It is the signed volume of the [[parallelepiped]] with edges '''a''', '''b''' and '''c''' and as such the vectors can be used in any order that's an [[even permutation]] of the above ordering. The following therefore are equal:

:<math>\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}), </math>

The [[vector triple product]] is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula

:<math>\begin{align}
\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) \\
(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \mathbf{b}(\mathbf{c} \cdot \mathbf{a}) - \mathbf{a} (\mathbf{b} \cdot \mathbf{c})
\end{align}</math>

The [[mnemonic]] "BAC minus CAB" is used to remember the order of the vectors in the right hand member. This formula is used in [[physics]] to simplify vector calculations. A special case, regarding [[gradient]]s and useful in [[vector calculus]], is
:<math>\begin{align}
  \nabla \times (\nabla \times \mathbf{f}) &= \nabla (\nabla \cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\
                                           &= \nabla (\nabla \cdot \mathbf{f} ) - \nabla^2 \mathbf{f},\\
\end{align}</math>

where ∇<sup>2</sup> is the [[vector Laplacian]] operator.

Other identities relate the cross product to the scalar triple product:
:<math>\begin{align}
(\mathbf{a}\times \mathbf{b})\times (\mathbf{a}\times \mathbf{c}) &= (\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})) \mathbf{a} \\
(\mathbf{a}\times \mathbf{b})\cdot(\mathbf{c}\times \mathbf{d}) &= \mathbf{b}^\mathrm{T} \left( \left( \mathbf{c}^\mathrm{T} \mathbf{a}\right)I - \mathbf{c} \mathbf{a}^\mathrm{T} \right) \mathbf{d}\\ &= (\mathbf{a}\cdot \mathbf{c})(\mathbf{b}\cdot \mathbf{d})-(\mathbf{a}\cdot \mathbf{d}) (\mathbf{b}\cdot \mathbf{c})
\end{align}</math>

where ''I'' is the identity matrix.