Question

A plane flies with a velocity of 52 m/s east through a12 m/s cross wind blowing the plane south. Find the magnitude and direction (relative to due east) of the resultant velocity at which it travels.

Answer 1

Just set up the vectors representing the forces (velocities, etc) acting on the plane. To add up the forces, just place each vector's base at the top of the previous vector. The final force is the vector from the base of the 1st force, to the tip of the last force.

In this case, start out at (0,0) and draw a vector pointing east of length 52. So, its tip is at (52,0)

Starting at (52,0), draw a vector pointing south of length 12. Now its tip is at (52,-12)

The resultant velocity is a vector from (0,0) to (52,-12). This is the hypotenuse of a right triangle with legs of length 52,12.
So, the final speed is 53.37 m/s.
The direction is the angle a with tan(a) = -12/52 = 13 deg south of east.