A standard related related rates question is:
A ladder propped aganist a wall is being moved closer. If the distance between the wall and the ladder reduces by reduces by $a$ m/s, find the rate of change in the height.
Set up an equation using the Pythagorean Theorem, and differentiate with respect to time: $$x^2+y^2=l^2 \implies 2x\dot{x}+2y\dot{y}=0$$
Substitute $\dot{x}=-a$ and solve for $\dot{y}$ $$2x(-a)+2y\dot{y}=0$$ $$\dot{y}=a\frac{x}{y}=a \text{ cot } \alpha $$
However, can this be solved using Vector methods. I know that $$\vec{v}_x=(-a,0)$$ But how do I proceed from there?