Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors with $\theta$ as the acute angle between them. If…
Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors with $\theta$ as the acute angle between them. If $\frac{1}{2}|\mathbf{a}-\mathbf{b}|=\sin \lambda \theta$, then $4 \lambda^2=$
4
1
3
2
Solution
It is given, that $\frac{1}{2}|\mathbf{a}-\mathbf{b}|=\sin (\lambda \theta)$
On squaring both sides, we get
$
\begin{aligned}
& \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2-2|\mathbf{a}||\mathbf{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 \quad \lambda=\frac{1}{2} \quad\left[\because(1-\cos \theta)=2 \sin ^2 \frac{\theta}{2}\right] \\
& \therefore \quad 4 \lambda^2=4\left(\frac{1}{2}\right)^2=1
\end{aligned}
$
Hence, option (b) is correct