Q250 / 270 · NDA 2022 (Phase 1) · Mathematics

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $|\vec{a}-\vec{b}| \lt 2$. If $2 \theta$ is the angle between $\vec{a}$ and $\vec{b}$, then which one of the following is correct?

  • A
    $0 \lt \sin \theta \lt 1$ only
  • B
    $-\frac{1}{2} \lt \sin \theta \lt \frac{1}{2}$ only
  • C
    $-1 \lt \sin \theta \lt 0$ only
  • Correct answer
    D
    $-1 \lt \sin \theta \lt 1$

Solution

Given that $|\vec{a}|=|\vec{b}|=1$ and $|\vec{a}-\vec{b}| \lt 2$ $\begin{aligned} & \Rightarrow|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^2 \lt 4 \\ & \Rightarrow|\overrightarrow{\mathrm{a}}|^2+|\overrightarrow{\mathrm{b}}|^2-2|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos 2 \theta \lt 4 \\ & \Rightarrow 1+1-2 \cos 2 \theta \lt 4 \\ & \Rightarrow \cos 2 \theta \gt -1 \\ & \Rightarrow 1-2 \sin ^2 \theta \gt -1 \\ & \Rightarrow \sin ^2 \theta \lt 1 \\ & \Rightarrow-1 \lt \sin \theta \lt 1\end{aligned}$

Hence, the correct answer is option: 'D'.

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