Dot Product of Vector Cross Products

Theorem
Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:

where $\left({\mathbf e_1, \mathbf e_2, \mathbf e_3}\right)$ is the standard ordered basis of $\mathbf V$.

Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.

Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$.

Then:


 * $\left({\mathbf a \times \mathbf b}\right) \cdot \left({\mathbf c \times \mathbf d}\right) = \left({\mathbf a \cdot \mathbf c}\right) \left({\mathbf b \cdot \mathbf d}\right) - \left({\mathbf a \cdot \mathbf d}\right) \left({\mathbf b \cdot \mathbf c}\right)$

Proof
Hence the result.