Question

Suppose u, v $\in$ V and ||u|| = ||v|| = 1 with < u,v > = 1: Prove that u = v.

Answer 1

Any two or more vectors will be equal if they are collinear, codirected, and have the same magnitude.

Given

$$||u||=||v||=1$$

$\langle u, v\rangle=1=>||u||\cdot||v||\cdot\cos\angle(u, v)=1$

$=>\cos\angle(u, v)=1=>\angle(u, v)=0$

$=>$ $u$ and $v$ are collinear, codirected.

The unit vectors $u$ and $v$ have the same magnitude, are collinear, codirected.

Therefore the vectors $u$ and $v$ are equal.