Q224 / 270 · NDA 2016 (Phase 1) · Mathematics

Let $\hat{a}, \hat{b}$ be two unit vectors and $\theta$ be the angle between them. What is $\cos \left(\frac{\theta}{2}\right)$ equal to?

  • A
    $\frac{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}{2}$
  • Correct answer
    B
    $\frac{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}{2}$
  • C
    $\frac{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}{4}$
  • D
    $\frac{|\hat{a}+\hat{b}|}{4}$

Solution

$\vec{a}$ and $\vec{b}$ are two unit vectors. $\because$ Hence, $\begin{aligned} |\vec{a}+\vec{b}|^2 & =(\hat{a}+\hat{b}) \cdot(\hat{a}+\hat{b}) \\ & =\hat{a} \cdot \hat{a}+\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{a}+\hat{b} \cdot \hat{b} \\ & =\hat{a} \cdot \hat{a}+\hat{b} \cdot \hat{b}+2 \hat{a} \cdot \hat{b} \\ & =|\hat{a}|^2+2|\hat{a}||\hat{b}| \cos \theta+|\hat{b} \| \hat{b}| \\ & =1+2 \cos \theta+1 \quad[\because \hat{b}, \hat{a} \text { are unit vector }] \\ & =2(1+\cos \theta) \\ |\hat{a}+\hat{b}|^2 & =2 \cdot 2 \cos ^2 \frac{\theta}{2} \\ \cos \frac{\theta}{2}= & \frac{|\hat{a}+\hat{b}|}{2} \end{aligned}$

Hence, the correct answer is option: 'B'.

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