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Q.
Two vectors $A$ and $B$ have equal magnitudes. If magnitude of $A + B$ is equal to $n$ times the magnitude of $A - B$, then the angle between $A$ and $B$ is
Let $\theta$ be the angle between $A$ and $B$
$| A + B |=n| A - B | $
$\Rightarrow \sqrt{A^{2}+B^{2}+2 A B \cos \theta} $
$=n \sqrt{A^{2}+B^{2}+2 A B \cos (180-\theta)} $
$\Rightarrow | A |-| B |-A-B-x$
$\Rightarrow 2 x^{2}(1+\cos \theta)=n^{2} \cdot 2 x^{2}(1-\cos \theta) $
$1+\cos \theta=n^{2}-n^{2} \cos \theta$
$\left(1+n^{2}\right) \cos \theta=n^{2}-1 $
$\cos \theta=\frac{n^{2}-1}{n^{2}+1} $
$\Rightarrow \theta=\cos ^{-1}\left(\frac{n^{2}-1}{n^{2}+1}\right)$