Vector Cross Product is Anticommutative

Theorem
The vector cross product is anticommutative:


 * $\forall \mathbf a, \mathbf b \in \R^3: \mathbf a \times \mathbf b = -\left({\mathbf b \times \mathbf a}\right)$

Complex Plane
The same result holds for complex numbers:

Proof 2
We start from a little lemma:

Lemma
Let $\mathbf x$ be a vector in a vector space of $3$ dimensions:


 * $\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$

Then $\mathbf x \times \mathbf x = 0$.

Proof of Lemma
Returning to the main proof, consider: