Vector Cross Product of Vector Cross Products

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

\(\displaystyle \mathbf a\) \(=\) \(\displaystyle a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3\)
\(\displaystyle \mathbf b\) \(=\) \(\displaystyle b_1 \mathbf e_1 + b_2 \mathbf e_2 + b_3 \mathbf e_3\)
\(\displaystyle \mathbf c\) \(=\) \(\displaystyle c_1 \mathbf e_1 + c_2 \mathbf e_2 + c_3 \mathbf e_3\)
\(\displaystyle \mathbf d\) \(=\) \(\displaystyle d_1 \mathbf e_1 + d_2 \mathbf e_2 + d_3 \mathbf e_3\)

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 $\mathbf b$.


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