Displacement exercise

The diagram looks like:

We want to find the length of the vector sum and its direction. In the diagram, all angles are measured in degrees. Notice that the angle marked with the purple arc is 180-30-45=105 degrees. Using the cosine rule:

D^{2} = D_{1}^{2} + D_{2}^{2} - 2D_{1}D_{2}cos(105°
) = (1 km)^{2} + (2 km)^{2} - 2(1 km)(2 km)cos(105°
)

D = 2.46 km

Then we find the direction by finding the angle marked with the orange arc. Let's call it q .

sin(105° )/D = sin(30° +q )/D2

So

Sin(30° + q ) = (2/2.26)sin105° = 0.7853

Thus:

30° + q = 0.90 rad = 52°

So the displacement is 2.46 km , 22° degrees east of north.