Lagrange's Formula

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Theorem

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


Then:

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


Corollary

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


Proof

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be embedded in a Cartesian $3$-space:

$\mathbf a = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}$, $\mathbf b = \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}$, $\mathbf c = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}$


\(\ds \mathbf b \times \mathbf c\) \(=\) \(\ds \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix} \times \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix}\)
\(\ds \) \(=\) \(\ds \begin {bmatrix} b_y c_z - b_z c_y \\ b_z c_x - b_x c_z \\ b_x c_y - b_y c_x \end {bmatrix}\) Definition of Vector Cross Product
\(\ds \mathbf a \times \paren {\mathbf b \times \mathbf c}\) \(=\) \(\ds \begin {bmatrix} a_x \\ a_y \\ a_z \end {bmatrix} \times \begin {bmatrix} b_y c_z - b_z c_y \\ b_z c_x - b_x c_z \\ b_x c_y - b_y c_x \end {bmatrix}\)
\(\ds \) \(=\) \(\ds \begin {bmatrix} a_y b_x c_y - a_y b_y c_x - a_z b_z c_x + a_z b_x c_z \\ a_z b_y c_z - a_z b_z c_y - a_x b_x c_y + a_x b_y c_x \\ a_x b_z c_x - a_x b_x c_z - a_y b_y c_z + a_y b_z c_y \end {bmatrix}\) Definition of Vector Cross Product
\(\ds \) \(=\) \(\ds \begin {bmatrix} a_y b_x c_y - a_y b_y c_x - a_z b_z c_x + a_z b_x c_z + a_x b_x c_x - a_x b_x c_x \\ a_z b_y c_z - a_z b_z c_y - a_x b_x c_y + a_x b_y c_x + a_y b_y c_y - a_y b_y c_y \\ a_x b_z c_x - a_x b_x c_z - a_y b_y c_z + a_y b_z c_y + a_z b_z c_z - a_z b_z c_z \end {bmatrix}\) adding $0 = a_i b_i c_i - a_i b_i c_i$ to each entry
\(\ds \) \(=\) \(\ds \begin {bmatrix} b_x \paren {a_y c_y + a_z c_z + a_x c_x} - c_x \paren {a_y b_y + a_z b_z + a_x b_x} \\ b_y \paren {a_z c_z + a_x c_x + a_y c_y} - c_y \paren {a_z b_z + a_x b_x + a_y c_y} \\ b_z \paren {a_x c_x + a_y c_y + a_z c_z} - c_z \paren {a_x b_x + a_y b_y + a_z c_z} \end {bmatrix}\)
\(\ds \) \(=\) \(\ds \begin {bmatrix} b_x \paren {\mathbf a \cdot \mathbf c} - c_x \paren {\mathbf a \cdot \mathbf b} \\ b_y \paren {\mathbf a \cdot \mathbf c} - c_y \paren {\mathbf a \cdot \mathbf b} \\ b_z \paren {\mathbf a \cdot \mathbf c} - c_z \paren {\mathbf a \cdot \mathbf b} \end {bmatrix}\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \paren {\mathbf a \cdot \mathbf c} \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix} - \paren {\mathbf a \cdot \mathbf b} \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix}\)
\(\ds \) \(=\) \(\ds \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf a \cdot \mathbf b} \mathbf c\)

$\blacksquare$


Source of Name

This entry was named for Joseph Louis Lagrange.


Sources

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