Vector Cross Product is Anticommutative/Proof 2

Theorem

The vector cross product is anticommutative:

$\forall \mathbf a, \mathbf b \in \R^3: \mathbf a \times \mathbf b = -\paren {\mathbf b \times \mathbf a}$

Proof

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

\(=\)

\(\ds \mathbf a \times \mathbf a + \mathbf a \times \mathbf b + \mathbf b \times \mathbf a + \mathbf b \times \mathbf b\)

Vector Cross Product Operator is Bilinear

\(\ds 0\)

\(=\)

\(\ds 0 + \mathbf a \times \mathbf b + \mathbf b \times \mathbf a + 0\)

Cross Product of Vector with Itself is Zero

\(\ds \mathbf a \times \mathbf b\)

\(=\)

\(\ds -\paren {\mathbf b \times \mathbf a}\)

simplifying

$\blacksquare$