Question

If a line makes an angle of \( 30^\circ \) with the positive direction of x-axis, \( 120^\circ \) with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:

• A. \( 90^\circ \)
• B. \( 120^\circ \)
• C. \( 60^\circ \)
• D. \( 0^\circ \)

Hint:

Remember that the direction cosines satisfy \( l^2 + m^2 + n^2 = 1 \) and correspond to the cosines of angles with the axes.

Answer 1

The Correct Option is A

Step 1: {Use the direction cosine property}
The direction cosines \( l, m, n \) of a line satisfy: \[ l^2 + m^2 + n^2 = 1. \] Here: \[ l = \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad m = \cos 120^\circ = -\frac{1}{2}. \] Step 2: {Solve for \( n \)}
Substitute \( l^2 \) and \( m^2 \) into the equation: \[ \left( \frac{\sqrt{3}}{2} \right)^2 + \left( -\frac{1}{2} \right)^2 + n^2 = 1. \] \[ \frac{3}{4} + \frac{1}{4} + n^2 = 1 \quad \Rightarrow \quad n^2 = 1 - 1 = 0. \] Thus, \( n = 0 \). Conclusion: The angle with the z-axis is \( 90^\circ \).