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Question
If \[\vec{a} \text{ and }\vec{b}\] be two unit vectors and θ the angle between them, then \[\vec{a} + \vec{b}\] is a unit vector if θ =
Options
(a) \[\frac{\pi}{4}\]
(b) \[\frac{\pi}{3}\]
(c) \[\frac{\pi}{2}\]
(d) \[\frac{2\pi}{3}\]
Solution
(d) \[\frac{2\pi}{3}\]
\[\text{ We have }\]
\[\left| \vec{a} \right| = 1 \text{ and } \left| \vec{b} \right| = 1\]
\[\text{ Now }, \left| \vec{a} + \vec{b} \right| = 1\]
\[ \Rightarrow \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2 \vec{a} . \vec{b} = 1\]
\[ \Rightarrow 1 + 1 + 2 \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 1\]
\[ \Rightarrow 2 + 2 \cos \theta = 1\]
\[ \Rightarrow 2 \cos \theta = - 1\]
\[ \Rightarrow \cos \theta = \frac{- 1}{2}\]
\[ \Rightarrow \theta = \frac{2\pi}{3}\]
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