Direction of umbrella

Let us consider the following example to understand how to find the direction in which the umbrella should be held; when the velocity of rain as well as the person is given.

Q. Rain is falling vertically with speed of 30 m/s. A woman rides a bicycle with a spedd of 10 m/s in N to S direction. In which direction should she hold the umbrella?

A. Let v_r (vector) be the velocity of rain and v (vector) be the velocity of woman. To the woman the rain will appear to come from the direction of v_r - v (vector) . Therefore, the woman should hold the umbrella in the direction of 

 v_r - v (vector) which is given by θ .

tan θ = v / v_r   = 10 / 30 = 1/3

i.e. she should hold the umbrella at an angle θ = tan ^-1 ( 1/3) south of vertical.