In studying the derivation for the general formula of a rotation about a directed line in $\Bbb R^3$, I came across an example of the reflection about the z-axis, and I have a rather simple question. Why does a rotation about the $z$ axis fix the vector $e_3 = (0,0,1)$ ? This is stated without explanation and it seems semi-intuitive but any further insight would be much appreciated.
2026-05-15 20:15:16.1778876116
Why does a rotation about the $z$-axis fix the vector $e_3 = (0,0,1)$
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The vector $e_3 = (0,0,1)$ forms a basis for the $z$-axis. Rotating about the $z$-axis is akin to rotating about the span of this vector so clearly it is held fixed. You can think about this visually as looking from above on the $\mathbb{R}^2$ plane. Objects in the plane get rotated as you rotate about the $z$-axis but the single point at the origin, which is projection of $e_3$ onto $\mathbb{R}^2$ remains unchanged.