Question

Can a vector have direction angles 45°, 60°, 120°?

Answer 1

Yes,
Let a vector makes an angle
\[\alpha = 45^{\circ} , \beta = 60^{\circ} , \gamma = 120^{\circ}\] with OX, OY, OZ respectively.
Let l, m, n be the direction cosines of the vector.
Then,
\[l = \cos 45^{\circ} = \frac{1}{\sqrt{2}} , m = \cos 60^{\circ} = \frac{1}{2} , n = \cos 120^{\circ} = - \frac{1}{2}\]
So,
\[l^2 + m^2 + n^2 = \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1\] 
Since, the vector has direction cosines such that
\[l^2 + m^2 + n^2 = 1\]
Hence, a vector can have direction angles \[45^{\circ}, 60^{\circ}, 120^{\circ}\]