Question

If a line makes angles of ${60}^{\circ }$ and ${45}^{\circ }$ with the positive directions of the $x$-axis and $y$-axis respectively, then the acute angle between the line and the $z$-axis is

• A. ${60}^{\circ }$
• B. ${45}^{\circ }$
• C. ${75}^{\circ }$
• D. ${15}^{\circ }$

Answer 1

Answer: (A)

${60}^{\circ }$

Let the angle with $z$ axis be $\gamma$
And as we know that
${\cos }^2$$\alpha$ $+$ ${\cos }^2$$\beta$ $+$ ${\cos }^2$$\gamma$ $=$ $1$
${\cos }^2$$60$ $+$ ${\cos }^2$45 $+$ ${\cos }^2$$\gamma$ $=$ $1$
$\frac{1}{4}$ $+$ $\frac{1}{2}$ $+$ ${\cos }^2$$\gamma$ $=$ $1$
${\cos }^2$$\gamma$ $=$ $1-$ $\frac{1}{4}$ $-$ $\frac{1}{2}$
${\cos }^2$$\gamma$ $=$ $\frac{1}{4}$
Since the angle is acute,
$\gamma$ $={60}^o$