Question

A line making angles $45^\circ$. and $60^\circ$ with the positive directions of the axis of $x$ and $y$, makes with the positive direction of $z$-axis, an angle of

• A. $60^\circ$
• B. $120^\circ$
• C. $60^\circ$ and $120^\circ$
• D. $None\, of \,these$

Answer 1

Answer: (A)

$60^\circ$

Answer 2

Let line make an angle $x$ with positive direction of z-axis .
Then, we have
$\cos^2 45^? + \cos^2 60^? + \cos^2 x = 1$
$\Rightarrow \left(\frac{1}{\sqrt{2}}\right)^{2} +\left(\frac{1}{2}\right)^{1}+\cos^{2} x = 1$
$\Rightarrow \:\:\: \cos^{2} x = 1 - \frac{1}{2} -\frac{1}{4}$
$\Rightarrow \:\:\: \cos^{2} \, x = \frac{1}{4} \:\: \Rightarrow \:\:\: \cos\, x = \frac{1}{2}$
(Neglecting $\cos\, x = \frac{-1}{2}$ because line is in I-octant)
$\Rightarrow \:\: x = 60^\circ$