Q218 / 270 · NDA 2018 (Phase 2) · Mathematics

What is $(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})$ equal to?

  • A
    $\overrightarrow{0}$
  • B
    $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$
  • Correct answer
    C
    $2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})$
  • D
    $|\overrightarrow{\mathrm{a}}|^2-|\overrightarrow{\mathrm{b}}|^2$

Solution

$\begin{aligned} & (\vec{a}-\vec{b}) \times(\vec{a}+\vec{b}) \\ = & \vec{a} \times \vec{a}+\vec{a} \times \vec{b}-\vec{b} \times \vec{a}-\vec{b} \times \vec{b} \\ = & 0+\vec{a} \times \vec{b}+\vec{a} \times \vec{b}-0 \\ = & 2(\vec{a} \times \vec{b})\end{aligned}$

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

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