A boat travels across a wide river. If the boat can travel at a speed of 15 mph and its pilot heads in a direction of 130.0 degrees, while the currents velocity is 2.1 mph at a heading of 200 degrees, what will be the final velocity of the boat?

The velocities of the boat relative to water v_1 and of the water relative to the banks v_2 can be expressed as vectors: v_1 = (v_1*cos theta_1; v_1*sin theta_1) = (15* cos 130.0°; 15* sin 130.0°) = (- 9.64; 11.49) and

v_2 = (v_2*cos theta_2; v_2*sin theta_2) = (2.1 * cos 200°; 2.1 * sin 130.0°) = (- 1.97; - 0.72).

The final velocity of the boat relative to the banks is

v = v_1 + v_2 = = (- 9.64+(-1.97); 11.49+(-0.72)) = (-11.6; 10.8), where

the final speed is v = sqrt{11.6^2 + 10.8^2} = 15.8 mph

and the direction is theta = 180° + tan^{-1} (10.8/-11/6) = 137°.