Q50 / 90 · JEE Main 2010 · Mathematics

A line $A B$ in three-dimensional space makes angles $45^{\circ}$ and $120^{\circ}$ with the positive $x$-axis and the positive $y$-axis respectively. If $A B$ makes an acute angle $\theta$ with the positive $z$-axis, then $\theta$ equals

  • A
    $45^{\circ}$
  • Correct answer
    B
    $60^{\circ}$
  • C
    $75^{\circ}$
  • D
    $30^{\circ}$

Solution

$$ \begin{aligned} & \ell=\cos 45^{\circ}=\frac{1}{\sqrt{2}} \\ & m=\cos 120^{\circ}=-\frac{1}{2} \\ & n=\cos \theta \end{aligned} $$ where $\theta$ is the angle which line makes with positive z-axis. Now $\ell^2+m^2+n^2=1$ $$ \begin{aligned} & \Rightarrow \frac{1}{2}+\frac{1}{4}+\cos ^2 \theta=1 \\ & \cos ^2 \theta=\frac{1}{4} \\ & \Rightarrow \cos \theta=\frac{1}{2} ( $\theta$ Being acute)\\ & \Rightarrow \theta=\frac{\pi}{3} \end{aligned} $$

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

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