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If ˆ a and ˆ b are unit vectors inclined at an angle θ, prove that - CBSE (Science) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

Question

If  $\hat{ a } \text{ and } \hat{b }$ are unit vectors inclined at an angle θ, prove that

$\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}$

Solution

$\text{ Given that } \hat{a}\text{ and } \hat{b}\text{ are unit vectors }.$

$So,\left| \hat{a} \right|=1,\left| \hat{b} \right|=1$

$\text{ We have }$

$\left| \hat{a} + \hat{b} \right|^2 = \left| \hat{a} \right|^2 + \left| \hat{b} \right|^2 + 2 \hat{a} .\hat{ b}$

$= 1 + 1 + 2 \left| \hat{a} \right| \left| \hat{b} \right| \cos \theta$

$= 2 + 2\cos \theta$

$\Rightarrow \cos\theta = \frac{\left| \hat{a} + \hat{b} \right|^2 - 2}{2} . . . \left( 1 \right)$

$\left| \hat{a} - b \right|^2 = \left| \hat{a} \right|^2 + \left| \hat{b} \right|^2 - 2 \hat{a} . \hat{b}$

$= 1 + 1 - 2 \left| \hat{a} \right| \left| \hat{b} \right| \cos \theta$

$= 2 - 2\cos \theta$

$\Rightarrow \cos\theta = \frac{2 - \left| \hat{a} - \hat{b} \right|^2}{2} . . . \left( 2 \right)$

$\sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}}$

$= \sqrt{\frac{1 - \frac{2 - \left| \hat{a} - \hat{b} \right|^2}{2}}{2}}[\text{ From } (2)]$

$= \sqrt{\frac{2 + \left| \hat{a} - \hat{b} \right|^2 - 2}{4}}$

$= \sqrt{\frac{\left| \hat{a} - \hat{b} \right|^2}{4}}$

$= \frac{1}{2}\left| \hat{a} - \hat{b} \right|$

$\text{ Now },$

$\tan \frac{\theta}{2} = \frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}} = \frac{\frac{1}{2}\left| \hat{a} - \hat{b} \right|}{\frac{1}{2}\left| \hat{a} + \hat{b} \right|} = \frac{\left| \hat{a} - \hat{b} \right|}{\left| \hat{a} + b \right|}$

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Solution If ˆ a and ˆ b are unit vectors inclined at an angle θ, prove that Concept: Basic Concepts of Vector Algebra.
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