- this shows that the cross product of two vectors transforms exactly like any other vector under a reference frame rotation
- orthonormal natural basis (also called standard basis):
$$
\bf e_1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad \bf e_2 = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \quad \cdots, \quad \bf e_n = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}
$$
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💡 Note that in 3-space, these would be orthonormal vectors $\bf e_1, \bf e_2, \bf e_3$
</aside>
- the definition of the cross product implies that the natural basis vectors satisfy the relation $\bf e_3 = \bf e_1 \times \bf e_2$ → can choose these vectors to obey the right hand rule
<aside>
💡 Since the natural basis vectors satisfy the right-handed coordinate system, cross products will obey the right hand rule if restricted to right handed reference frames.
</aside>
- note that if a transformation matrix is an improper orthogonal matrix with determinant -1, it would change a right-handed reference frame into a left-handed one