Discussion
In spherical polar coordinates \((r, \theta, \phi)\), the unit vector \(\hat{\theta}\) at \((10, \pi/4, \pi/2)\) is
Answer: D
The unit vector is \(\frac{1}{\sqrt{2}}(\hat{j} - \hat{k})\).
In Cartesian components,
\[ \begin{aligned} \hat{\theta} &= \cos\theta\cos\phi\,\hat{i} + \cos\theta\sin\phi\,\hat{j} \\ &\quad - \sin\theta\,\hat{k} \end{aligned} \]
which does not depend on \(r\). At \(\theta = \pi/4\), \(\phi = \pi/2\): \(\cos\phi = 0\), \(\sin\phi = 1\), and \(\cos\theta = \sin\theta = 1/\sqrt{2}\), so
\[ \hat{\theta} = \tfrac{1}{\sqrt{2}}(\hat{j} - \hat{k}) \]
\(\hat{\theta}\) points in the direction of increasing polar angle, away from the +z axis, so its z-component is negative. \((\hat{j} + \hat{k})/\sqrt{2}\) is \(\hat{r}\) at that point; \((-\hat{j} + \hat{k})/\sqrt{2}\) is \(-\hat{\theta}\); and \(\hat{k}\) is wrong in direction altogether.
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