Question

Find $|\vec{a}|$ and $|\vec{b}|$ ,if $(\vec{a}+\vec{b}).(\vec{a}-\vec{b})=8$ and $|\vec{a}|=8|\vec{b}|.$

Answer 1

The correct answer is: $\frac{16\sqrt{2}}{3\sqrt{7}}.$
$(\vec{a}.\vec{b}).(\vec{a}-\vec{b})=8$
$⇒\vec{a}.\vec{a}-\vec{a}.\vec{b}+\vec{b}.\vec{a}-\vec{b}.\vec{b}=8$
$⇒|\vec{a}|^2-|\vec{b}|^2=8$
$⇒(8|\vec{b}|)^2-|\vec{b}|^2=8 [|\vec{a}|=8|\vec{b}|]$
$⇒64|\vec{b}|^2-|\vec{b}|^2=8$
$⇒63|\vec{b}|^2=8$
$⇒|\vec{b}|^2=\frac{8}{63}$
$⇒|\vec{b}|^=\sqrt{\frac{8}{63}}$ [Magnitude of a vector is non-negative]
$⇒|\vec{b}|=\frac{2\sqrt{2}}{3\sqrt{7}}$
$⇒|\vec{a}|=8|\vec{b}|=\frac{8×2\sqrt{2}}{3\sqrt{7}}=\frac{16\sqrt{2}}{3\sqrt{7}}.$