# How to calculate velocity of the ship in the direction of the current?

A ship travels with velocity given by [1 2], with current flowing in the direction given by [1 1​] with respect to some co-ordinate axes.

What is the velocity of the ship in the direction of the current?

Since the problem is using velocity as a vector, we can continue that. If you want speed, the take the absolute value of the answer. We want the projection of the ship velocity onto the current velocity. Let $u=(1,2)$ the ship. Let $v=(1,1)$, the current. $$v\cdot \left ( \frac{u\cdot v}{|v|^2} \right )=(1.5,1.5)$$ Intuitively, we want to divide the ships velocity into x and y parts where the x part is in the direction of the current and the y part is 90 degrees to the current. You could do that any number of ways, but the easiest way to write notation, is the one shown.