Step-by-Step Solution
**Step 1: Recall the Direction Cosines Formula**
The sum of the squares of the direction cosines of a line is equal to 1. If a line makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the x, y, and z axes respectively, then:
\[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]
**Step 2: Substitute the given angles**
We are given that \(\alpha = 30^{\circ}\) and \(\beta = 120^{\circ}\). We need to find \(\gamma\). Substituting the given values into the formula:
\[\cos^2 (30^{\circ}) + \cos^2 (120^{\circ}) + \cos^2 \gamma = 1\]
**Step 3: Evaluate the cosine values**
We know that \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\) and \(\cos(120^{\circ}) = -\frac{1}{2}\). Substituting these values:
\[\left(\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 + \cos^2 \gamma = 1\]
\[\frac{3}{4} + \frac{1}{4} + \cos^2 \gamma = 1\]
**Step 4: Simplify the equation**
\[\frac{4}{4} + \cos^2 \gamma = 1\]
\[1 + \cos^2 \gamma = 1\]
**Step 5: Solve for \(\cos^2 \gamma\)**
\[\cos^2 \gamma = 1 - 1\]
\[\cos^2 \gamma = 0\]
**Step 6: Solve for \(\gamma\)**
\[\cos \gamma = 0\]
\[\gamma = \arccos(0)\]
\[\gamma = 90^{\circ}\]
Correct Answer: \(90^{\circ}\)<\/strong>