Rain is falling vertically with a speed of $30 m{s}^{-1}$. A woman rides a bicycle with a $12 m{s}^{-1}$ speed in east to west direction. In which direction she should hold her umbrella?

In figure $\overrightarrow{v_r}$ represents the velocity of rain and $\overrightarrow{v_b}$ the velocity of the bicycle, the woman is riding. To protect herself from rain, the woman should hold her umbrella in the direction of relative velocity of rain with respect to the bicycle ${\overrightarrow{v}}_{rb}$.

From figure, 

$\tan \theta =\frac{v_b}{v_r}=\frac{12}{30}=\frac{2}{5} or \theta ={\tan }^{-1}\left(\frac{2}{5}\right)$

Therefore, the woman should hold her umbrella at an angle of ${\tan }^{-1}\left(\frac{2}{5}\right)$ with the vertical towards the west.