Q231 / 270 · NDA 2021 (Phase 2) · Mathematics

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be unit vectors such that $\vec{a} \times \vec{b}$ is perpendicular to $\vec{c}$. If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then which of the following is/are correct? 1. $\vec{a} \times \vec{b}=\sin \theta \vec{c} \quad 2 . \quad \vec{a} \cdot(\vec{b} \times \vec{c})=0$ Select the correct answer using the code given below.

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

Solution

Since $\vec{a} \times \vec{b}$ is perpendicular to $\therefore(\vec{a} \times \vec{b}) \cdot \vec{c}=0=\vec{a} \cdot(\vec{b} \times \vec{c})$ Let $\vec{a} \times \vec{b}=\sin \theta \vec{c}$ $\begin{aligned} & \Rightarrow(\vec{a} \times \vec{b}) \cdot \vec{c}=\sin \theta \vec{c} \cdot \vec{c}=\sin \theta|\vec{c}| \\ & \Rightarrow 0=\sin \theta \Rightarrow \theta=0, \pi \end{aligned}$ So it is possible only when $\vec{a}$ and $\vec{b}$ are parallel or collinear. Hence, statement 1 is wrong.

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

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