Vector Cross Product Distributes over Addition/Proof 1

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:

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

\(=\)

\(\ds \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix} \times \paren {\begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix} + \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix} }\)

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix} \times {\begin {bmatrix} b_x + c_x \\ b_y + c_y \\ b_z + c_z \end {bmatrix} }\)

Definition of Vector Sum

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} a_y \paren {b_z + c_z} - a_z \paren {b_y + c_y} \\ a_z \paren {b_x + c_x} - a_x \paren {b_z + c_z} \\ a_x \paren {b_y + c_y} - a_y \paren {b_x + c_x} \end {bmatrix}\)

Definition of Vector Cross Product

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} a_y b_z + a_y c_z - a_z b_y - a_z c_y \\ a_z b_x + a_z c_x - a_x b_z - a_x c_z \\ a_x b_y + a_x c_y - a_y b_x - a_y c_x \end {bmatrix}\)

Real Multiplication Distributes over Addition

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} a_y b_z - a_z b_y + a_y c_z - a_z c_y \\ a_z b_x - a_x b_z + a_z c_x - a_x c_z \\ a_x b_y - a_y b_x + a_x c_y - a_y c_x \end {bmatrix}\)

Real Addition is Commutative

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end {bmatrix} + \begin {bmatrix} a_y c_z - a_z c_y \\ a_z c_x - a_x c_z \\ a_x c_y - a_y c_x \end {bmatrix}\)

Definition of Vector Sum

\(\ds \)

\(=\)

\(\ds \paren {\begin {bmatrix}a_x \\ a_y \\ a_z \end {bmatrix} \times \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix} } + \paren {\begin {bmatrix} a_x \\ a_y \\ a_z \end {bmatrix} \times \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix} }\)

Definition of Vector Cross Product

\(\ds \)

\(=\)

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

$\blacksquare$