Q96 / 180 · COMEDK 2024 (Morning Shift) · Mathematics

The vector $(\vec{r})$ whose magnitude is $3 \sqrt{2}$ units which makes an angle of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{2}$ with $y$ and $z$- axis respectively is

  • A
    $-\hat{\imath} \pm \hat{\jmath}$
  • B
    $\hat{\imath} \pm 3 \hat{\jmath}$
  • C
    $\hat{\imath} \pm \hat{\jmath}$
  • Correct answer
    D
    $\pm 3 \hat{\imath}+3 \hat{\jmath}$

Solution

Direction cosines: $m = \cos\dfrac{\pi}{4} = \dfrac{1}{\sqrt{2}}$, $n = \cos\dfrac{\pi}{2} = 0$

Using $l^2 + m^2 + n^2 = 1$:
$l^2 + \dfrac{1}{2} + 0 = 1 \Rightarrow l = \pm\dfrac{1}{\sqrt{2}}$

$\vec{r} = |\vec{r}|(l\hat{i} + m\hat{j} + n\hat{k}) = 3\sqrt{2}\left(\pm\dfrac{1}{\sqrt{2}}\hat{i} + \dfrac{1}{\sqrt{2}}\hat{j} + 0\hat{k}\right) = \pm 3\hat{i} + 3\hat{j}$

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

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