True/false? $v,w$ are vectors in $\mathbb{R}^2$. The vector $u=\left \langle v,w \right \rangle v -\left \| v \right \|^2w$ lies vertically on $v$

Question

True/false? $v,w$ are vectors in $\mathbb{R}^2$. The vector $u=\left \langle v,w \right \rangle v -\left \| v \right \|^2w$ lies vertically on $v.$

Can you please explain me how a task like that is solved? It's not homework but a task from an old exercise (you want see it?).

I know that two vectors $a,b$ to be orthogonal we need that $\left \langle a,b \right \rangle=0$

So applying that to this task, we need that $\left \langle u,v \right \rangle=0$, is this correct?

I think this will not be zero if we don't have zero vector, so statement is false? Because it's not satisfied for all vectors I think.

But please I need explanation and how to form things if it's required here at all.

Answer 1

$$\langle v,u\rangle=\langle v,\,\langle v,w\rangle v-\left\|v\right\|^2 w\rangle=\langle v,w\rangle\langle v,v\rangle-\left\|v\right\|^2\langle v,w\rangle=0\implies v\perp u$$