A unit vector ` vec r` makes angle `pi/3\ a n d\ pi/2` with ` hat j a n d\ hat k` respectively and an acute angle `theta\ w i t h\ i ,` Find `thetadot`

Explanation

To solve the problem, we need to find the acute angle \( \theta \) that the unit vector \( \vec{r} \) makes with the \( \hat{i} \) axis, given that it makes angles \( \frac{\pi}{3} \) (or \( 60^\circ \)) with the \( \hat{j} \) axis and \( \frac{\pi}{2} \) (or \( 90^\circ \)) with the \( \hat{k} \) axis.

### Step-by-Step Solution:

1. **Understanding the Unit Vector:**
A unit vector \( \vec{r} \) can be expressed in terms of its direction cosines:
\[
\vec{r} = l \hat{i} + m \hat{j} + n \hat{k}
...