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# If ^ a and ^ B Are Unit Vectors Inclined at an Angle θ, Prove that Cos θ 2 = 1 2 ∣ ∣ ^ a + ^ B ∣ ∣ - CBSE (Arts) 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 $\cos\frac{\theta}{2} = \frac{1}{2}\left| \hat{a} + \hat{b} \right|$

#### Solution

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

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

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

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

$= 2 + 2\cos \theta$

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

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

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

$= 2 - 2\cos \theta$

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

$\text{ Now },$

$\cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}}$

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

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

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

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

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Solution If ^ a and ^ B Are Unit Vectors Inclined at an Angle θ, Prove that Cos θ 2 = 1 2 ∣ ∣ ^ a + ^ B ∣ ∣ Concept: Basic Concepts of Vector Algebra.
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