Vector Cross Product of Vector Cross Products/Corollary

Theorem

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

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

Then:

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

Proof

From Vector Cross Product of Vector Cross Products:

\(\ds \paren {\mathbf a \times \mathbf b} \times \paren {\mathbf c \times \mathbf d}\)

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \leadsto \ \ \)

\(\ds \sqbrk {\mathbf a, \mathbf b, \mathbf c} \mathbf d\)

\(=\)

\(\ds \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c - \sqbrk {\mathbf a, \mathbf c, \mathbf d} \mathbf b + \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a\)

rearranging

\(\ds \)

\(=\)

\(\ds \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a - \sqbrk {\mathbf a, \mathbf c, \mathbf d} \mathbf b + \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c\)

rearranging

\(\ds \)

\(=\)

\(\ds \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a + \sqbrk {\mathbf a, \mathbf d, \mathbf c} \mathbf b + \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c\)

Equivalent Expressions for Scalar Triple Product

$\blacksquare$

Sources

• 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 6$: $(16)$