Q223 / 270 · NDA 2021 (Phase 1) · Mathematics

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|=4$, then which one of the following is correct?

  • A
    $\vec{a}$ and $\vec{b}$ must be unit vectors.
  • B
    $\vec{a}$ must be parallel to $\vec{b}$.
  • Correct answer
    C
    $\vec{a}$ must be perpendicular to $\vec{b}$.
  • D
    $\vec{a}$ must be equal to $\vec{b}$.

Solution

$|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$ On squaring both sides, we get $|\vec{a}+\vec{b}|^2=|\vec{a}-\vec{b}|^2$ $|\vec{a}|^2+|\vec{b}|^2+2 \vec{a} \cdot \vec{b}=|\vec{a}|^2+|\vec{b}|^2-2 \vec{a} \cdot \vec{b}$ $\vec{a} \cdot \vec{b}=0 \quad \therefore \quad \vec{a} \perp \vec{b}$

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

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