Q101 / 180 · COMEDK 2023 (Morning Shift) · Mathematics

$\mathrm{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \mathrm{b}=\hat{\mathrm{i}}-\hat{\mathrm{j}}$ and $\mathrm{c}=5 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$, then unit vector parallel to $\mathrm{a}+\mathrm{b}-\mathrm{c}$ but in opposite direction is

  • A
    $\dfrac{1}{3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$
  • Correct answer
    B
    $\dfrac{1}{3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$
  • C
    None of these
  • D
    $\dfrac{1}{2}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$

Solution

Given vectors $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = \hat{i} - \hat{j}$, and $\vec{c} = 5\hat{i} - \hat{j} + \hat{k}$.

Calculate the vector $\vec{v} = \vec{a} + \vec{b} - \vec{c}$:
$\vec{v} = (2\hat{i} + \hat{j} - \hat{k}) + (\hat{i} - \hat{j}) - (5\hat{i} - \hat{j} + \hat{k})$
$\vec{v} = (2 + 1 - 5)\hat{i} + (1 - 1 + 1)\hat{j} + (-1 - 0 - 1)\hat{k}$
$\vec{v} = -2\hat{i} + \hat{j} - 2\hat{k}$

The magnitude of $\vec{v}$ is $|\vec{v}| = \sqrt{(-2)^2 + (1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$.

The unit vector in the direction of $\vec{v}$ is $\hat{u} = \dfrac{\vec{v}}{|\vec{v}|} = \dfrac{1}{3}(-2\hat{i} + \hat{j} - 2\hat{k})$.

The unit vector in the opposite direction is $-\hat{u} = -\dfrac{1}{3}(-2\hat{i} + \hat{j} - 2\hat{k}) = \dfrac{1}{3}(2\hat{i} - \hat{j} + 2\hat{k})$.

Answer: $\dfrac{1}{3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$

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

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