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Q.
If the sum of two unit vectors is a unit vector, then magnitude of difference is
Motion in a Plane
Solution:
Let $\hat{n}_{1}$ and $\hat{n}_{2}$ are the two unit vectors, then the sum is
$\vec{n}_{s}=\hat{n}_{1}+\hat{n}_{2} $
or $n_{s}^{2} =n_{1}^{2}+n_{2}^{2}+2 n_{1} n_{2} \cos \theta $
$=1+1+2 \cos \theta$
Since it is given that $n_{s}$ is also a unit vector, therefore
$ 1=1+1+2 \cos \theta $
$\Rightarrow \cos \theta=-\frac{1}{2}$
$\therefore \theta=120^{\circ}$
Now the difference vector is $ \hat{n}_{d}=\hat{n}_{1}-\hat{n}_{2}$
or $ n_{d}^{2}=n_{1}^{2}+n_{2}^{2}-2 n_{1} n_{2} \cos \theta=1+1-2 \cos \left(120^{\circ}\right)$
$\therefore n_{d}^{2}=2-2(-1 / 2)=2+1=3 $
$\Rightarrow n_{d}=\sqrt{3}$