If $U$ and $V$ are unit vectors and $U\cdot V=1$, is it true that $U=V$?

Question

EDIT: without using $U\cdot V=\lVert U\rVert \lVert V\rVert\cos{\theta}$,

If $U$ and $V$ are unit vectors and $U\cdot V=1$, does that mean $U=V$? I know $U\cdot V=U\cdot U=V\cdot V=1$. But I also know $U\cdot V=U\cdot U$ doesn't necessarily mean $V=U$.

I can also see:

$U\cdot (V-U)=0$ and $-V\cdot(V-U)=0$ So $U$ and $V$ are perpendicular to the same vector. I'm not sure where to go from here.

Answer 1

Calculate $(U-V)\cdot(U-V)$ and use $|x|=0\implies x=0$.

Answer 2

Since generally $U\cdot V=\lVert U\rVert \lVert V\rVert \cos\theta$, where $\theta$ is the angle between $U$ and $V$,

we have $1=1\cdot1\cdot\cos\theta$,

so $\cos\theta=1$, so...