Lagrange's Formula/Corollary

Theorem

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

Then:

$\paren {\mathbf a \times \mathbf b} \times \mathbf c = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf b \cdot \mathbf c} \mathbf a$

$\ds \mathbf c \times \paren {\mathbf a \times \mathbf b}$

$=$

$\ds \paren {\mathbf c \cdot \mathbf b} \mathbf a - \paren {\mathbf c \cdot \mathbf a} \mathbf b$

$\ds \leadsto \ \ $

$\ds \paren {\mathbf a \times \mathbf b} \times \mathbf c$

$=$

$\ds -\paren {\paren {\mathbf c \cdot \mathbf b} \mathbf a - \paren {\mathbf c \cdot \mathbf a} \mathbf b}$

$\ds $

$=$

$\ds -\paren {\mathbf b \cdot \mathbf c} \mathbf a + \paren {\mathbf a \cdot \mathbf c} \mathbf b$

$\ds $

$=$

$\ds \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf b \cdot \mathbf c} \mathbf a$

$\blacksquare$

This entry was named for Joseph Louis Lagrange.

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving Dot and Cross Products: $22.19$