In the undergraduate text of physics, the equation for velocity and displacement at constant acceleration are given in scalar form. For example, my text reads
$$v^2 = v_0^2 + 2a(x-x_0)$$
But today, I am reading another text which gives the vector form for displacement. I am trying to write above equation in vector form also but how do I deal with that square in vector? I am tackling that square with dot product of two vector such that $v^2 \to \vec{v}\cdot\vec{v}$, is that correct? So what about the multiplication of acceleration and the displacement? dot product again?
This equation looks a bit confusing to me. Let's say I throw a stone upright with initial speed $v_0$ and it reaches the highest point sometimes later (so $v=0$), and the coordinate is given with y axis down and x axis horizontal. Note that the displacement is negative since the coordinate's yaxis is downward, also the gravity is along the positive y axis so the equation becomes $$ 0 = v_0^2 + 2g(-x) $$ which gives $$ x = \frac{v_0^2}{2g} $$ is that right?