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If $\hat{a}$ and $\hat{b}$ are unit vectors inclined at an angle $\theta$ ,than prove that cos$\Large\frac{\theta}{2}$=$\Large\frac{1}{2}$$|\hat{a}$+$\hat{b}| $

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  • For any two vectors $ \hat a \: and \: \hat b$ $(\hat a + \hat b)^2=(\hat a)^2+2\hat a.\hat b+(\hat b)^2=a^2+2\hat a.\hat b+b^2$ $(\hat a-\hat b)^2=a^2-2\hat a.\hat b+b^2$ $(\hat a+\hat b).(\hat a-\hat b)=a^2-b^2$
Step 1
$ \hat a\: and \: \hat b$ are unit vectors.
$ \therefore |\hat a|=|\hat b|=1$
$ (\hat a+\hat b)^2=a^2+2\hat a.\hat b+b^2 = 1+2(1)(1) \cos\theta+1$
$ = 2(1+ \cos\theta)$
$ = 2.2 \cos^2 \large\frac{\theta}{2}$
Step 2
$ (\hat a+\hat b)^2 = |\hat a+\hat b|^2=4 \cos^2 \large\frac{\theta}{2}$
$ \therefore |\hat a+\hat b| = 2 \cos \large\frac{\theta}{2}$
answered Jun 1, 2013 by thanvigandhi_1

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