Geometrical Derivation of the Polar Unit Vector?

Question

Does anyone know what is wrong with my derivation of the polar unit vector, $\vec{e}_{\theta}$? My Derivation does not match Wikipedia's Result. Alternatively, do you know how to derive the polar unit vector in spherical coordinates from a geometrical standpoint?

Answer 1

For $\vec e_\theta$ in terms of $\vec e_z$ and your $\vec e_w,$

$$ \vec e_\theta = (\cos \theta) \vec e_w - (\sin\theta) \vec e_z. $$

What you wrote for this step is incorrect, as you should be able to verify by taking $\theta$ near $0.$ In that case $\cos\theta \approx 1$ and $\sin\theta \approx 0,$ so your formula would result in $\vec e_\theta \approx -\vec e_z.$ Should $\vec e_\theta$ point almost straight down when $\theta$ is very small?

The formula in Wikipedia is consistent with the formula in this answer.

Answer 2

Your problem is in the very first step. You got your $\sin \theta$ and $\cos \theta$ swapped.