107. u and v are two non-collinear unit vectors such that | u + v 2 + u × v |=1 . Then the value of | u × v | is equal to

BITSAT 2022

Exam: BITSAT
Year: 2022
Subject: Mathematics
Chapter: Vector Algebra

BITSAT 2022 · Mathematics · Vector Algebra

107. u and v are two non-collinear unit vectors such that | u + v 2...

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

Options

A $\left|\frac{\hat{\mathbf{u}}+\hat{\mathbf{v}}}{2}\right|$
B $|\hat{\mathbf{u}}+\hat{\mathbf{v}}|$
C $|\hat{\mathbf{u}}-\hat{\mathbf{v}}|$
D $\left|\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{2}\right|$

Correct answer

D. $\left|\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{2}\right|$

Step-by-step solution

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}=\cos2 \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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