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107. $\hat{\mathbf{u}}$ and $\hat{\mathbf{v}}$ are two non-collinear unit vectors such that $\left|\frac{\hat{\mathbf{u}}+\hat{\mathbf{v}}}{\mathbf{2}}+\hat{\mathbf{u}} \times \hat{\mathbf{v}}\right|=1$. Then the value of $|\hat{\mathbf{u}} \times \hat{\mathbf{v}}|$ is equal to
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Verified Answer
The correct answer is:
$\left|\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{2}\right|$
$\left|\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{2}\right|$
Given that, $\left|\frac{\hat{\mathbf{u}}+\hat{\mathbf{v}}}{2}+\hat{\mathbf{u}} \times \hat{\mathbf{v}}\right|=1$
$$
\begin{array}{ll}
\Rightarrow & \left|\frac{\hat{\mathbf{u}}+\hat{\mathbf{v}}}{2}+\hat{\mathbf{u}} \times \hat{\mathbf{v}}\right|^2=1 \\
\Rightarrow & \frac{2+2 \cos \theta}{4}+\sin ^2 \theta=1 \\
& \quad[\because \hat{\mathbf{u}} \cdot(\hat{\mathbf{u}} \times \hat{\mathbf{v}})=\hat{\mathbf{v}}(\hat{\mathbf{u}} \times \hat{\mathbf{v}})=0] \\
\Rightarrow & \cos ^2 \frac{\theta}{2}=\cos 2 \theta \\
\Rightarrow & \theta=n \pi \pm \frac{\theta}{2}, n \in z \\
\Rightarrow & \theta=\frac{2 \pi}{3}
\end{array}
$$
$\Rightarrow \quad|\hat{\mathbf{u}} \times \hat{\mathbf{v}}|=\sin \frac{2 \pi}{3}=\sin \frac{\pi}{3}=\left|\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{2}\right|$
$$
\begin{array}{ll}
\Rightarrow & \left|\frac{\hat{\mathbf{u}}+\hat{\mathbf{v}}}{2}+\hat{\mathbf{u}} \times \hat{\mathbf{v}}\right|^2=1 \\
\Rightarrow & \frac{2+2 \cos \theta}{4}+\sin ^2 \theta=1 \\
& \quad[\because \hat{\mathbf{u}} \cdot(\hat{\mathbf{u}} \times \hat{\mathbf{v}})=\hat{\mathbf{v}}(\hat{\mathbf{u}} \times \hat{\mathbf{v}})=0] \\
\Rightarrow & \cos ^2 \frac{\theta}{2}=\cos 2 \theta \\
\Rightarrow & \theta=n \pi \pm \frac{\theta}{2}, n \in z \\
\Rightarrow & \theta=\frac{2 \pi}{3}
\end{array}
$$
$\Rightarrow \quad|\hat{\mathbf{u}} \times \hat{\mathbf{v}}|=\sin \frac{2 \pi}{3}=\sin \frac{\pi}{3}=\left|\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{2}\right|$
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