Question

Can a line have direction angles as ${45}^o,{45}^o$ & ${60}^o$.

Answer 1

No, a line can't have the given direction angles.

Proof by contradiction:-

If possible let there exists a line which has the direction angles ${45}^o,{45}^o,{60}^o$.

Then its direction cosines will be $(l,m,n)=(\cos {45}^o,\cos {45}^o,\cos {60}^o)=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{2})$.

The from the property of d.cs we must have

$l^2+m^2+n^2=1$.

But in this case

$l^2+m^2+n^2$

$=\frac{5}{4}\ne 1$.

So our assumption is wrong, there can't be any line having the given direction angles.