If a line makes an angle of 60°, 135°, 120° with the positive x, y, z-axis, respectively, then find the direction cosines.

Correct option is A

​$$\begin{aligned}&\text{The direction cosines } (l, m, n) \text{ of a line are the cosines of the angles } (\alpha, \beta, \gamma) \text{ that the line makes with} \\&\text{the positive } x-, y-, \text{ and } z\text{-axes, respectively.} \\&\qquad l = \cos \alpha, \quad m = \cos \beta, \quad n = \cos \gamma \\\\&\textbf{Compute Each Direction Cosine} \\&\text{Given:} \\&\qquad \bullet \ \text{Angle with the } x\text{-axis } (\alpha) = 60^\circ \\&\qquad \bullet \ \text{Angle with the } y\text{-axis } (\beta) = 135^\circ \\&\qquad \bullet \ \text{Angle with the } z\text{-axis } (\gamma) = 120^\circ \\&\text{Calculate the direction cosines:} \\&\qquad l = \cos 60^\circ = \frac{1}{2} \\&\qquad m = \cos 135^\circ = \cos(180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{1}{\sqrt{2}} \\&\qquad n = \cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2} \\\\&\textbf{Verify the Direction Cosines} \\&\text{The sum of the squares of the direction cosines should equal 1:} \\&\qquad l^2 + m^2 + n^2 = \left(\frac{1}{2}\right)^2 + \left(-\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 = \frac{1}{4} + \frac{1}{2} + \frac{1}{4} = \frac{1 + 2 + 1}{4} = 1\end{aligned}$$​​