Vector Cross Product of Vector Cross Products

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Theorem

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

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

Let $\sqbrk {\mathbf a, \mathbf b, \mathbf c}$ denote the scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$.


Then:

\(\ds \paren {\mathbf a \times \mathbf b} \times \paren {\mathbf c \times \mathbf d}\) \(=\) \(\ds \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c - \sqbrk {\mathbf a, \mathbf b, \mathbf c} \mathbf d\)
\(\ds \) \(=\) \(\ds \sqbrk {\mathbf a, \mathbf c, \mathbf d} \mathbf b - \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a\)


Corollary

$\sqbrk {\mathbf a, \mathbf b, \mathbf c} \mathbf d = \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a + \sqbrk {\mathbf a, \mathbf d, \mathbf c} \mathbf b + \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c$


Proof

\(\ds \paren {\mathbf a \times \mathbf b} \times \paren {\mathbf c \times \mathbf d}\) \(=\) \(\ds \paren {\paren {\mathbf a \times \mathbf b} \cdot \mathbf d} \mathbf c - \paren {\paren {\mathbf a \times \mathbf b} \cdot \mathbf c} \mathbf d\) Lagrange's Formula
\(\ds \) \(=\) \(\ds \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c - \sqbrk {\mathbf a, \mathbf b, \mathbf c} \mathbf d\) Definition of Scalar Triple Product


Then:

\(\ds \paren {\mathbf a \times \mathbf b} \times \paren {\mathbf c \times \mathbf d}\) \(=\) \(\ds \paren {\mathbf a \cdot \paren {\mathbf c \times \mathbf d} } \mathbf b - \paren {\mathbf b \cdot \paren {\mathbf c \times \mathbf d} } \mathbf a\) Lagrange's Formula: Corollary
\(\ds \) \(=\) \(\ds \sqbrk {\mathbf a, \mathbf c, \mathbf d} \mathbf b - \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a\) Definition of Scalar Triple Product

$\blacksquare$


Sources

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