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Question
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors inclined at an angle θ, then the value of \[\left| \vec{a} - \vec{b} \right|\]
Options
(a) \[2 \sin\frac{\theta}{2}\]
(b) 2 sin θ
(c) \[2 \cos\frac{\theta}{2}\]
(d) 2 cos θ
Solution
(a) \[2 \sin\frac{\theta}{2}\]
\[\vec{a} . \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ = 1 \times 1 \cos \theta(\text{ Because } \vec{a} \text{ and } \vec{b} \text{ are unit vectors })\]
\[ = \cos \theta . . . \left( i \right)\]
\[ \left| \vec{a} - \vec{b} \right|^2 = \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - 2 \vec{a} . \vec{b} \]
\[ = 1 + 1 - 2 \cos \theta \left[ \text{ Using } \left( i \right) \right]\]
\[ = 2 - 2 \cos \theta \]
\[ = 2 \left( 1 - \cos \theta \right)\]
\[ = 2 \left( 2 \sin^2 \frac{\theta}{2} \right)\]
\[ = 4 \sin^2 \frac{\theta}{2}\]
\[ \therefore \left| \vec{a} - \vec{b} \right| = 2 \sin \frac{\theta}{2}\]
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