Given two non-zero vectors a and b such that |a + b| = |a - b|, find the value of a • b

Question

Given two non-zero vectors $\vec{a}$ and $\vec{b}$ such that |$\vec{a}$ + $\vec{b}$| = |$\vec{a}$ - $\vec{b}$|, find the value of $\vec{a}$ • $\vec{b}$.

Answer 1

$|\vec{a} + \vec{b}|^{2} = |\vec{a} - \vec{b}|^{2}$
$\vec{a}+\vec{a}\cdot\vec{b}+ \vec{b}=\vec{a}-\vec{a}\cdot\vec{b}+ \vec{b} \\\vec{a}\cdot\vec{b}=0 $

Answer 2

HINT: $$||a+b||=||a-b|| \implies ||a+b||^2=||a-b||^2$$ which is just $$(a+b)\cdot(a+b)=(a-b)\cdot(a-b)$$ Expand that equation and see how things cancel.

Answer 3

It is equivalent to

$$ |a+b|^2 = |a-b|^2 $$

and s o

$$ (a+b) \cdot (a+b) = (a-b) \cdot (a-b) $$

and so

$$ a^2 + 2 a \cdot b + b^2 = a^2 - 2 a \cdot b + b^2 $$

can you finish it ?