For nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, is the length of the projection of $v$ along $u$ always less than the length of $v$.

Question

For all nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, the length of the projection of $v$ along $u$ is less than the length of $v$.

This is a true/false question and when I said true it was marked as wrong. Now I am looking for some help/clarification.

I've tried this two different ways but I need to justify my reasoning. If I choose $v=[13;6]$ and $u= [16;0]$ then it holds true because my projection is only $[13;0]$.

Thinking what if my u was shorter I picked a vector of $[12;0]$ which then gave me a projection of $[13;0]$ again. Is there something I am missing here?

Answer 1

It could be equal; consider for example if $v$ is a multiple of $u$.