Question

You’re traveling 45 km at 30o north east, then 98 km at 50o north west. What is your resultant displacement?

Answer 1

Let's first find the $x$- and $y$- components of the resultant displacement:

$$d_x=45\ km\times cos30^{\circ}+98\ km\times cos(180^{\circ}-50^{\circ})=-24\ km,$$ $$d_y=45\ km\times sin30^{\circ}+98\ km\times sin(180^{\circ}-50^{\circ})=97.6\ km.$$

We can find the magnitude of the resultant displacement from the Pythagorean theorem:

$$d=\sqrt{d_x^2+d_y^2}=\sqrt{(-24\ km)^2+(97.6\ km)^2}=100.5\ km.$$

We can find the direction of the resultant displacement from the geometry:

$$\theta=tan^{-1}(\dfrac{d_y}{d_x}),$$ $$\theta=tan^{-1}(\dfrac{97.6\ km}{-24\ km})=76.2^{\circ} N\ of\ W.$$