Question

If θ be the angle between the unit vectors a and b, then $\cos\frac{\theta}{2} =$

• A. $\frac{1}{2}|\mathbf{a} - \mathbf{b}|$
• B. $\frac{1}{2}|\mathbf{a} + \mathbf{b}|$
• C. $\frac{|\mathbf{a} - \mathbf{b}|}{|\mathbf{a} + \mathbf{b}|}$
• D. $\frac{|\mathbf{a} + \mathbf{b}|}{|\mathbf{a} - \mathbf{b}|}$

Answer 1

Answer: (B)

$\frac{1}{2}|\mathbf{a} + \mathbf{b}|$

Answer 2

$(\mathbf{a} + \mathbf{b}).(\mathbf{a} + \mathbf{b}) = |\mathbf{a}|^{2} + |\mathbf{b}|^{2} + 2\mathbf{a}.\mathbf{b}$

or $|\mathbf{a} + \mathbf{b}|^{2} = 2.2\cos^{2}\frac{\theta}{2} \Rightarrow \cos\frac{\theta}{2} = \frac{1}{2}|\mathbf{a} + \mathbf{b}|.$