Question

If $\vec{a}$ and $\vec{b}$ are any two perpendicular vectors of equal magnitude such that $|3\vec{a}+4\vec{b}|+|4\vec{a}-3\vec{b}|=20$, then $|\vec{a}|$ equals ________.

Answer 1

Answer: 2

Let $|\vec{a}|=|\vec{b}|=\lambda$.
Also, $\vec{a}\cdot \vec{b}=0$ (Given)
$|3\vec{a}+4\vec{b}{|}^2=9{\lambda }^2+16{\lambda }^2=25{\lambda }^2$ ....(1)
And $|4\vec{a}-3\vec{b}{|}^2=16{\lambda }^2+9{\lambda }^2=25{\lambda }^2$ ......(2)
$\therefore |3\vec{a}+4\vec{b}|+|4\vec{a}-3\vec{b}|=5\lambda +5\lambda =20$ (Given)
So, $\lambda =2=|\vec{a}|=|\vec{b}|$