Angles of vectors of same magnitude

Question

$\vec{a} $ and $ \vec{b}$ are equal in magnitude. What is the angle between $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$?

I'm aware it is $ 90$, but I don't know how to demonstrate it.

Answer 1

Hint: $\require{cancel}\;(\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b}) = \vec{a}\cdot\vec{a}-\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{a}-\vec{b}\cdot\vec{b}= \cancel{|a|^2} - \bcancel{\vec{a} \cdot \vec{b}}+\bcancel{\vec{a}\cdot\vec{b}}- \cancel{|b|^2}=\cdots\,$

Or, geometrically, think at a certain rhombus and recall that the diagonals are orthogonal.