homework and exercises - Relative motion involving angle

Suppose I am going in a direction with a velocity $v_1$ and my friend is going in a direction which makes an angle of $A$ with my direction with a velocity of $v_2$.

Then what will be my relative motion with respect to my friend or his relative motion with respect to me?

Every book I have read gave example where two things are going parallel or opposite of me. But I've never found any example with an angle.

Answer

Let's say your velocity is $\vec{v}_1$ is

$$\vec{v}_1= v_{1x}\hat{i} + v_{1y}\hat{j}$$ and your friends velocity $\vec{v}_2$ is $$\vec{v}_2= v_{2x}\hat{i}+v_{2y}\hat{j}$$ Then your velocity relative to your friend will be $$\vec{v}_r = (v_{1x}-v_{2x})\hat{i} + (v_{1y}-v_{2y})\hat{j}$$ The components can be found by considering an appropriate coordinate system. In this case that will be a coordinate system with the $x$-axis aligned with your friends velocity. In that case $v_{2y} =0$. While the components of your velocity will be $v_{1x}=v_{1}\cos{A}$ and $v_{1y}=v_{1}\sin{A}$.

Hope this helps.