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
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 6$: $(15)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving Dot and Cross Products: $22.21$