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

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}|$