Q245 / 270 · NDA 2023 (Phase 2) · Mathematics

Consider the following in respect of the vectors $\vec{a}=(0,1,1)$ and $\vec{b}=(1,0,1)$ : 1. The number of unit vectors perpendicular to both $\vec{a}$ and $\vec{b}$ is only one. 2. The angle between the vectors is $\frac{\pi}{3}$. Which of the statement given above is/are correct?

  • A
    1 only
  • Correct answer
    B
    2 only
  • C
    both 1 and 2
  • D
    Neither 1 nor 2

Solution

Number of units vectors perpendiculars to $\vec{a}$ and $\vec{b}$ is two ie., $\frac{ \pm \vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$ and angle between $\vec{a}$ and $\vec{b}$ is $\begin{aligned} & \cos \theta=\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{~b}}}{|\mathrm{a}||\mathrm{b}|} \\ & \Rightarrow \cos \theta=\frac{(0,1,1) \cdot(1,0,1)}{\sqrt{2} \cdot \sqrt{2}}=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3} \end{aligned}$

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

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