Standard Basis Vectors

Standard basis vectors i, j, k and vector components of a, denoted here ax, ay, az

If $(\mathbf{i},\mathbf{j},\mathbf{k})$ is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities:

$$\begin{array}{rlrl}\mathbf{i}& \times \mathbf{j}& & =\mathbf{k}\\ \mathbf{j}& \times \mathbf{k}& & =\mathbf{i}\\ \mathbf{k}& \times \mathbf{i}& & =\mathbf{j}\end{array}$$

The anticommutativity of the Cross Product implies that:

$$\begin{array}{rlrl}\mathbf{j}& \times \mathbf{i}& & =-\mathbf{k}\\ \mathbf{k}& \times \mathbf{j}& & =-\mathbf{i}\\ \mathbf{i}& \times \mathbf{k}& & =-\mathbf{j}\end{array}$$

The anticommutativity of the cross product (and the obvious lack of Linear Independence) also implies that:

$$\mathbf{i}\times \mathbf{i}=\mathbf{j}\times \mathbf{j}=\mathbf{k}\times \mathbf{k}=0$$