Question

If a line makes ${45}^o,{60}^o$ with positive direction of axes $x$ and $y$ then the angle it makes with the z-axis is:

• A. ${30}^o$
• B. ${90}^o$
• C. ${45}^o$
• D. ${60}^o$

Answer 1

Answer: (D)

${60}^o$

If $\alpha ,\beta ,\gamma$ are the angles made by a line with the positive direction of $x$-axis,$y$-axis and $z$-axis then $cos\alpha ,cos\beta ,cos\gamma$ are called the direction cosines of the line.

Also $co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$

Given $\alpha ={45}^{\circ },\beta ={60}^{\circ }$

$\Rightarrow co{s}^2{45}^{\circ }+co{s}^2{60}^{\circ }+co{s}^2\gamma =1$

$\Rightarrow {\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+co{s}^2\gamma =1$

$\Rightarrow \frac{1}{2}+\frac{1}{4}+co{s}^2\gamma =1$

$\Rightarrow co{s}^2\gamma =1-\frac{1}{2}-\frac{1}{4}$

$\Rightarrow co{s}^2\gamma =1-\frac{3}{4}$

$\Rightarrow co{s}^2\gamma =\frac{1}{4}$

$\Rightarrow cos\gamma =\frac{1}{2}$

$\Rightarrow \gamma =co{s}^{-1}\left(\frac{1}{2}\right)$

$\Rightarrow \gamma ={60}^{\circ }$