# Vector Cross 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:

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

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 $\mathbf b$.

Then:

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