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?
Can anyone please help me with this question with intuition behind it?
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.