Question

Show that $|a|b+|b|a$ is perpendicular to $|a|b-|b|a$ for any two non-zero vectors a and b.

Answer 1

Let $p=|a|b+|b|a$ and $q=|a|b-|b|a$

Then, $p.q=(|a|b+|b|a).(|a|b-|b|a)$

$=|a{|}^2(b.b)-|a||b|(b.a)+|b||a|(a.b)-|b{|}^2(a.a)$

$=|a{|}^2|b{|}^2-|a||b|(a.b)+|a||b|(a.b)-|b{|}^2|a{|}^2=0$

$\Rightarrow p\perp q$ ($\because$ if c.d = 0 $\Rightarrow$ c is perpendicular to d)

Hence, |a| b + |b| a and |a| b - |b| a are perpendicular to each other for any two non-zero vectors a and b.