Question

Suppose u, v ∈ V and ||u|| = ||v|| = 1 with < u, v > = 1.

Prove that u = v.

Please assist.

Answer 1

Any two vectors $u$ and $v$ will be equal if they are collinear, codirected, and have the same magnitude.

Given $||u||=||v||=1, (u, v)=1.$

Then

$$(u,v)=||u||\cdot||v||\cdot \cos\widehat{(u,v)}$$

$$1=1\cdot1\cdot\cos \widehat{(u,v)}=>\cos \widehat{(u,v)}=1$$

$$=>\widehat{(u,v)}=0$$

If the angle between two non-zero vectors equal zero , then the vectors are parallel in the same direction.

We have two non-zero, collinear, codirected, unit vectors $u$ and $v.$

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

$$u=v$$