Q249 / 270 · NDA 2022 (Phase 1) · Mathematics

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that $\vec{a} \times \vec{b}=\vec{c}$. Consider the following statements : 1. $\vec{a}$ is unique if $\vec{b}$ and $\vec{c}$ are given 2. $\vec{c}$ is unique if $\vec{a}$ and $\vec{b}$ are given Which of the above statements is/are correct?

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

Solution

Given that $\vec{a} \times \vec{b}=\vec{c}$ $\Rightarrow \mathrm{a} \perp \overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{b}} \perp \overrightarrow{\mathrm{c}}$ If $\vec{b}$ and $\vec{c}$ are given then any vector $\vec{a}$ coplanar with $\overrightarrow{\mathrm{b}}$ is also perpendicular to $\overrightarrow{\mathrm{c}}$. So, $\vec{a}$ is not unique If $\vec{a}$ and $\vec{b}$ is given then its cross product is unique So $\overrightarrow{\mathrm{c}}$ is unique.

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

Relevant Questions