Q222 / 270 · NDA 2021 (Phase 1) · Mathematics

Consider the following statements in respect of a vector $\vec{c}=\vec{a}+\vec{b}$, where $|\vec{a}|=|\vec{b}| \neq 0$ : 1. $\vec{c}$ is perpendicular to $(\vec{a}-\vec{b})$. 2. $\vec{c}$ is perpendicular to $(\vec{a} \times \vec{b})$. Which of the above statements is/are correct?

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

Solution

$(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=|\vec{a}|^2-|\vec{b}|^2$ $=0 \quad(\because|\vec{a}|=|\vec{b}|)$ $\therefore$ Vector $\vec{c}=(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$. Also, $\vec{c} \cdot(\vec{a}-\vec{b})=\left[\begin{array}{lll}\vec{c} & \vec{a} & \vec{b}\end{array}\right]$ $\therefore \vec{c}$ is perpendicular to $(\vec{a} \times \vec{b})$.

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

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