Question

Let $a$ and $b$ be unit vectors with $\theta$ as the acute angle between them. If $\frac{1}{2}|a-b|=\sin \lambda \theta$, then $4{\lambda }^2=$

• A. 4
• B. 1
• C. 3
• D. 2

Answer 1

Answer: (B)

It is given, that $\frac{1}{2}|a-b|=\sin (\lambda \theta )$
On squaring both sides, we get
$\Rightarrow |a{|}^2+|b{|}^2-2|a||b|\cos \theta =4{\sin }^2(\lambda \theta )$
$\Rightarrow 1+1-2\cos \theta =4{\sin }^2(\lambda \theta )$
$\Rightarrow 2(1-\cos \theta )=4{\sin }^2(\lambda \theta )$
$\Rightarrow 4{\sin }^2\left(\frac{\theta }{2}\right)=4{\sin }^2(\lambda \theta )$
$\Rightarrow \lambda =\frac{1}{2}\left[\because (1-\cos \theta )=2{\sin }^2\frac{\theta }{2}\right]$
$\therefore 4{\lambda }^2=4{\left(\frac{1}{2}\right)}^2=1$