Question

The direction cosines of a line which lies in the \( ZOX \) plane and makes an angle of \( 30^\circ \) with the \( Z \)-axis are

• A. \( 0, \dfrac{1}{2}, \pm \dfrac{\sqrt{3}}{2} \)
• B. \( \pm \dfrac{1}{2}, 0, \dfrac{\sqrt{3}}{2} \)
• C. \( 0, \dfrac{\sqrt{3}}{2}, \pm \dfrac{1}{2} \)
• D. \( \dfrac{\sqrt{3}}{2}, 0, \pm \dfrac{1}{2} \)

Hint:

For lines lying in coordinate planes, one of the direction cosines is always zero.

Answer 1

The Correct Option is B

Step 1: Use the condition of the plane.
Since the line lies in the \( ZOX \) plane, the direction cosine along the \( y \)-axis is zero. \[ m = 0 \] Step 2: Use the angle with the \( Z \)-axis.
If the line makes an angle of \( 30^\circ \) with the \( Z \)-axis, then \[ n = \cos 30^\circ = \frac{\sqrt{3}}{2} \] Step 3: Use the relation between direction cosines.
\[ l^2 + m^2 + n^2 = 1 \] \[ l^2 + 0 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 \] Step 4: Solve for \( l \).
\[ l^2 = 1 - \frac{3}{4} = \frac{1}{4} \Rightarrow l = \pm \frac{1}{2} \] Step 5: Conclusion.
Hence, the direction cosines are \[ \left( \pm \frac{1}{2}, 0, \frac{\sqrt{3}}{2} \right) \]