A river flows in a direction from south west to north east at a velocity of 7.1 m/s. A boat captain wants to cross this river to reach a point on the opposite shore due east of the boat’s current position. The boat moves at 13 m/s. Which direction should it head towards if the resultant velocity is 19.74 m/s?

The river flows 45º N of E, so its velocity vector is

w = (7.1 m/s) (cos(45º) i + sin(45º) j) ≈ (5.02 m/s) (i + j)

The boat captain wants to move against the current such that the boat moves due E, so that the resultant is

r = (19.74 m/s) cos(0º) i = (19.74 m/s) i

The boat's speed in still water is 13 m/s, and should be moved in a direction θ such that the boat's velocity will be

b = (13 m/s) (cos(θ) i + sin(θ) j)

We need to have

w + b = r

Match up the components and solve for θ :

5.02 m/s + (13 m/s) cos(θ) = 19.74 m/s

5.02 m/s + (13 m/s) sin(θ) = 0

cos(θ) ≈ 1.13

sin(θ) ≈ -0.386

tan(θ) = sin(θ) / cos(θ) ≈ -0.341

θ ≈ -18.8º

So the captain should point the boat 18.8º S of E.