Question

Find |a| and |b|, if (a + b). (a - b) = 8 and |a| = 8 |b|.

Answer 1

Given, (a + b). (a - b) = 8 and |a| = 8 |b|
$\Rightarrow a.a-a.b+b.a-b.b=8$
$\Rightarrow |a{|}^2-|b{|}^2=8$ $(\because a.a=|a{|}^{2}anda.b=b.a)$
$\Rightarrow (8|b|{)}^2-|b{|}^2=8\Rightarrow 63|b{|}^2=8$ (Given, |a|=8|b|)
$\Rightarrow |\overrightarrow{b}|=\sqrt{\frac{8}{63}}=\frac{2}{3}\sqrt{\frac{2}{7}}$
Also, $|a|=8|b|=8\left(\frac{2}{3}\sqrt{\frac{2}{7}}\right)=\frac{16}{3}\sqrt{\frac{2}{7}}$