Q156 / 240 · KCET 2026 · Mathematics

If $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}, \vec{b} = \alpha\hat{i} + \beta\hat{j} + 2\hat{k}$ and $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then $\alpha + \beta$ is equal to

  • A
    $2$
  • B
    $-1$
  • C
    $0$
  • Correct answer
    D
    $1$

Solution

Given $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$

Squaring both sides:

$|\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b} = |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}$

$4\vec{a} \cdot \vec{b} = 0$

$\vec{a} \cdot \vec{b} = 0$

Substituting the given vectors $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = \alpha\hat{i} + \beta\hat{j} + 2\hat{k}$:

$(2)(\alpha) + (2)(\beta) + (-1)(2) = 0$

$2\alpha + 2\beta - 2 = 0$

$2(\alpha + \beta) = 2$

$\alpha + \beta = 1$

Answer: $1$

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

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