A line $A B$ in three dimensions makes angles $45^{\circ}$ and $120^{\circ}$ with the positive $X$-axis and…
A line $A B$ in three dimensions 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
$30^{\circ}$
$45^{\circ}$
$60^{\circ}$
$75^{\circ}$
Solution
Here, line $A B$ in three dimensions makes an angle $45^{\circ}$ and $120^{\circ}$ with positive $X$-axis and $Y$-axis respectively and acuts angle $\theta$ with positive $Z$-axis.
$\therefore \cos ^2 45^{\circ}+\cos ^2 120^{\circ}+\cos ^2 \theta=1$
$\Rightarrow \quad\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{2}\right)^2+\cos ^2 \theta=1$
$\Rightarrow \quad \cos ^2 \theta=1-\left(\frac{1}{2}+\frac{1}{4}\right)$
$\Rightarrow \quad \cos ^2 \theta=1 / 4 \Rightarrow \cos \theta=1 / 2$
$\Rightarrow \quad \theta=60^{\circ}$