First slide

Vectors

Question

If the sum of two unit vectors is a unit vector, then magnitude of difference is

Moderate

A

$\sqrt{2}$
B

$\sqrt{3}$
C

$\frac{1}{\sqrt{2}}$
D

$\sqrt{5}$

Solution

Let ${\hat{n}}_{1} and {\hat{n}}_{2}$ be the two unit vectors, then the sum is
$\bar{n} = {\hat{n}}_{1} + {\hat{n}}_{2}$
or $n_{s}^{2} = n_{1}^{2} + n_{2}^{2} + 2 n_{1} n_{2} cosθ$
= 1+1+2 $\cos θ$
since it is given that $n_{s}$ is also a unit vector, therefore
1 = 1+1+2cos $θ$ $\Rightarrow cosθ = - \frac{1}{2} ∴ θ = 120^{0}$
Now the difference vector is ${\hat{n}}_{d} = {\hat{n}}_{1} - {\hat{n}}_{2}$ or
$n_{d}^{2} = n_{1}^{2} + n_{2}^{2} - 2 n_{1} n_{2} cosθ = 1 + 1 - 2 \cos (120^{0})$
$∴ n_{d}^{2} = 2 - 2 (- \frac{1}{2}) = 2 + 1 = 3 \Rightarrow n_{d} = \sqrt{3}$

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