Q:

Vectors of same length, perpendicularity

How to algebraically show that if two vectors i.e. $\vec a$ and $\vec b$ have the same length then $\vec a+\vec b$ vector is perpendicular to $\vec a-\vec b$?
I get this graphically but cannot put it algebraically. 

A:

For vectors $a$ and $b$ to have the same length means that $\langle a,a\rangle=\langle b,b\rangle$. Now use bilinearity of the scalar product:
$$\langle a+b,a-b\rangle=\langle  a,a\rangle-\langle a,b\rangle+\langle b,a\rangle-\langle b,b\rangle=0$$.

