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:

Use the identity that the squares of the direction cosines add to 1, then pick the obtuse angle.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Direction cosines satisfy \(l^2+m^2+n^2=1\).

Step 2: Known angles
\(l=\cos45^{\circ}=\frac{1}{\sqrt2}\) and \(m=\cos60^{\circ}=\frac{1}{2}\).

Step 3: Find n
\(n^2=1-\frac{1}{2}-\frac{1}{4}=\frac{1}{4}\), so \(n=\pm\frac{1}{2}\).

Step 4: Obtuse angle
For an obtuse angle, \(\cos\theta=-\frac{1}{2}\), so \(\theta=120^{\circ}\).

Step 5: Compare the options
The angle is \(120^{\circ}\), the second option.

Final Answer:
The obtuse angle with the Z-axis is \(120^{\circ}\). \[ \boxed{\text{(B) }120^{\circ}} \]