Relative Motion in Two Dimensions

A boat crossing a river has two velocities at once: its velocity relative to the water (set by its engine or paddling) and the water's velocity relative to the ground (the current). What an observer on the bank sees is the sum of these two, added as vectors, not as plain numbers. If the boat aims straight across while the current pushes downstream, those two velocities are perpendicular, so they combine head-to-tail into a resultant arrow that points diagonally across and downstream. That resultant is the boat's actual path over the ground, which is why aiming straight across always lands you downstream of your starting point.

To add the vectors, draw the boat's velocity arrow, then start the current's velocity arrow at the tip of the first one. The resultant runs from the tail of the first arrow to the tip of the second. When the two velocities are perpendicular, the resultant's magnitude comes from the Pythagorean theorem, and its direction comes from the inverse tangent of one leg over the other. If instead you want to land exactly across, you tilt the boat's heading upstream so one component of its velocity cancels the current entirely, leaving only the across component to carry you over.

This idea is just vector addition applied to velocities, the same skill used for displacement and acceleration elsewhere in kinematics. It sets up projectile motion, where horizontal and vertical velocities are independent components that add the same way, and it reinforces that velocity, like displacement, has both size and direction.