Question

Show that number line in the space can make angle $\frac{\pi }{6}$ and $\frac{\pi }{4}$ which $X-$axis and $Y-$axis.

Answer 1

Let, if possible, a line in space make angles $\frac{\pi }{6}$ and $\frac{\pi }{4}$ with $X-$axis and $Y-$axis.

$\therefore \alpha =\frac{\pi }{6},\beta =\frac{\pi }{4}$

Let the line make angle $\gamma$ with $Z-$axis

$\because {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

$\because {\cos }^2{\left(\frac{\pi }{6}\right)}^2+{\cos }^2{\left(\frac{\pi }{4}\right)}^2+{\cos }^2\gamma =1$

$\because {\cos }^2{\left(\frac{\sqrt{3}}{2}\right)}^2+{\cos }^2{\left(\frac{2}{\sqrt{2}}\right)}^2+{\cos }^2\gamma =1$

$\therefore {\cos }^2\gamma =1-\frac{3}{4}-\frac{1}{2}=-\frac{1}{4}$

This is not possible, because $\cos \gamma$ is real.

$\therefore {\cos }^2\gamma$ cannot be negative.

Hence, there is no line in space which makes angles $\frac{\pi }{6}$ and $\frac{\pi }{4}$ with $X-$axis and $Y-$axis.