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) $
$\Rightarrow 4 \sin ^{2}\left(\frac{\theta}{2}\right)=4 \sin ^{2}(\lambda \theta)$
$\Rightarrow \lambda=\frac{1}{2} \,\,\,\left[\because(1-\cos \theta)=2 \sin ^{2} \frac{\theta}{2}\right]$
$\therefore 4 \lambda^{2}=4\left(\frac{1}{2}\right)^{2}=1$