Distance between 2 Orthogonal Unit vectors

Question

Is this true that the Distance between any 2 Orthogonal unit vectors in any inner product space is always equal to $\sqrt2$ ?

Answer 1

HINT

Let consider

• $u_1,\quad |u_1|=1\implies u_1\cdot u_1=|u_1|^2=1$

• $u_2,\quad |u_2|=1\implies u_2\cdot u_2=|u_2|^2=1$

• $u_1\cdot u_2=0$

then compute

• $d^2=(u_2-u_1)\cdot(u_2-u_1)$

Answer 2

Hint: If $u$ and $v$ are orthogonal and have norm $1$, what can you say about$$\|u-v\|=\sqrt{\langle u-v,u-v\rangle}?$$