Prove if the following vectors are orthonormal set

Question

Prove:

If u * v = 0 and |u||v| = 1, then it form an orthonormal set.

To prove the set is orthonormal set, I must prove that |u| = |v| = 1, But i’m stock, I was wondering if anyone can give me some advice.

Answer 1

Counterexample: in ${\mathbb R}^2$ take $$ x = (2,0), \qquad y=(0,{1\over2}) $$ Both of your conditions are satisfied, yet these vectors do not form an orthonormal set.

Answer 2

It is not true. Counterexample: $u = (2,0)$ and $v=(0,1/2)$.

Answer 3

Not necessarily true |u|=1/|v| is also OK