How to prove perpendicular vectors problem

Question

If I have: $$|\underline a + \underline b| = |\underline a - \underline b|$$ how do I prove that $\underline a$ is perpendicular to $\underline b$?

Answer 1

First we say norm of vector and not absolute value of vector. Second we have $$\langle a,b\rangle=\frac14(||a+b||^2-||a-b||^2)$$ which gives the desired result.

Answer 2

Same answer as given elsewhere really, but possibly in a more familiar notation: $$\|a+b\|^2=(a+b)\cdot(a+b)=a\cdot a+2a\cdot b+b\cdot b$$ and $$\ \|a-b\|^2=(a-b)\cdot(a-b)=a\cdot a-2a\cdot b+b\cdot b\ ;$$ since these are equal we have $2a\cdot b=-2a\cdot b$, so $a\cdot b=0$, so the vectors are perpendicular.