Vector Cross Product Distributes over Addition

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Theorem

The vector cross product is distributive over addition.

That is, in general:

$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$

for $\mathbf a, \mathbf b, \mathbf c \in \R^3$.


Proof

Let:

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

be vectors in $\R^3$.


Then:

\(\displaystyle \mathbf a \times \paren {\mathbf b + \mathbf c}\) \(=\) \(\displaystyle \begin{bmatrix} a_x \\ a_y \\a_z \end{bmatrix} \times \paren {\begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} + \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix} }\)
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_x \\ a_y \\a_z \end{bmatrix} \times {\begin{bmatrix} b_x + c_x \\ b_y + c_y \\ b_z + c_z \end{bmatrix} }\) Definition of Vector Addition
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_y \paren {b_z + c_z} - a_z \paren {b_y + c_y} \\ a_z \paren {b_x + c_x} - a_x \paren {b_z + c_z} \\ a_x \paren {b_y + c_y} - a_y \paren {b_x + c_x} \end{bmatrix}\) Definition of Vector Cross Product
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_y b_z + a_y c_z - a_z b_y - a_z c_y \\ a_z b_x + a_z c_x - a_x b_z - a_x c_z \\ a_x b_y + a_x c_y - a_y b_x - a_y c_x \end{bmatrix}\) Real Multiplication Distributes over Addition
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_y b_z - a_z b_y + a_y c_z - a_z c_y \\ a_z b_x - a_x b_z + a_z c_x - a_x c_z \\ a_x b_y - a_y b_x + a_x c_y - a_y c_x \end{bmatrix}\) Real Addition is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{bmatrix} + \begin{bmatrix} a_y c_z - a_z c_y \\ a_z c_x - a_x c_z \\ a_x c_y - a_y c_x \end{bmatrix}\) Definition of Vector Addition
\(\displaystyle \) \(=\) \(\displaystyle \paren {\begin{bmatrix}a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} } + \paren {\begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix} }\) Definition of Vector Cross Product
\(\displaystyle \) \(=\) \(\displaystyle \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}\)

$\blacksquare$


Sources