Scalars and Vectors

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Question

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

Moderate

Solution

Let ${\hat{n}}_{1} and {\hat{n}}_{2}$ are the two unit vectors, then the sum and difference of these two vectors be represented by
$\begin{array}{l} {\hat{n}}_{s} = {\hat{n}}_{1} + {\hat{n}}_{2} and {\hat{n}}_{d} = {\hat{n}}_{1} - {\hat{n}}_{2} \\ \Rightarrow n_{s}^{2} = n_{1}^{2} + n_{2}^{2} + 2 n_{1} n_{2} \cos θ = 1 + 1 + 2 \cos θ \end{array}$
Since it is given that n_s is also a unit vector, therefore we have
1 = 1+1+ 2 cos $θ$
$\begin{array}{l} \Rightarrow \cos θ = - \frac{1}{2} \\ \Rightarrow θ = 120 \end{array}$
The difference of the two vectors is given by
$\begin{array}{l} {\hat{n}}_{d} = {\hat{n}}_{1} - {\hat{n}}_{2} \\ \Rightarrow n_{d}^{2} = n_{1}^{2} + n_{2}^{2} - 2 n_{1} n_{2} \cos θ = 1 + 1 - 2 \cos (120^{o}) \\ \Rightarrow n_{d}^{2} = 2 - 2 (- \frac{1}{2}) = 2 + 1 = 3 \\ \Rightarrow n_{d} = \sqrt{3} = 1 .73 \end{array}$

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