If $\theta$ is the angle between the unit vectors $\mathbf{a}$ and $\mathbf{b}$, then $\sin…
If $\theta$ is the angle between the unit vectors $\mathbf{a}$ and $\mathbf{b}$, then $\sin \frac{\theta}{2}$ is equal to
- $\left|\frac{\mathbf{a}-\mathbf{b}}{\mathbf{a}+\mathbf{b}}\right|$
- $|\mathbf{a}+\mathbf{b}|$
- $|\mathbf{a}-\mathbf{b}|$
- $\frac{1}{2}|\mathbf{a}-\mathbf{b}|$
Solution
$\begin{aligned} & \text { Given, }|\mathbf{a}|=|\mathbf{b}|=1 \\ & |\mathbf{a}-\mathbf{b}|^2=|\mathbf{a}|^2+|\mathbf{b}|^2-2|\mathbf{a}||\mathbf{b}| \cos \theta \\ & \Rightarrow|\mathbf{a}-\mathbf{b}|^2=1+1-2 \cos \theta \\ & \Rightarrow \quad|\mathbf{a}-\mathbf{b}|^2=2(1-\cos \theta) \\ & \Rightarrow \quad|\mathbf{a}-\mathbf{b}|^2=2\left(2 \sin ^2\left(\frac{\theta}{2}\right)\right) \\ & \Rightarrow \quad \sin ^2 \frac{\theta}{2}=\frac{1}{4}|\mathbf{a}-\mathbf{b}|^2 \\ & \Rightarrow \quad \sin \frac{\theta}{2}=\frac{1}{2}|\mathbf{a}-\mathbf{b}| \\ & \end{aligned}$
Asked in: AP EAMCET 2021 (25 Aug Shift 1)
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