Question

If the directed line makes an angle $45^\circ$ and $60^\circ$ with the X and Y -axes respectively, then the obtuse angle $\theta$ made by the line with the Z -axis is

• A. $135^\circ$
• B. $120^\circ$
• C. $160^\circ$
• D. $150^\circ$

Hint:

Sum of squares of direction cosines = 1.

Answer 1

The Correct Option is A

Concept:
Direction cosines satisfy: \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \] Step 1: Substitute given angles. \[ \cos^2 45^\circ + \cos^2 60^\circ + \cos^2 \theta = 1 \] \[ \frac{1}{2} + \frac{1}{4} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = \frac{1}{4} \]
Step 2: Find angle. \[ \cos \theta = \pm \frac{1}{2} \] Obtuse angle: \[ \theta = 120^\circ \text{ or } 135^\circ \] From direction cosine condition → obtuse value: \[ \theta = 135^\circ \]
Step 3: Conclusion. \[ {135^\circ} \]