**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 \(\gamma\)** \[\cos \gamma = 0\] \[\gamma = \arccos(0)\] \[\gamma = 90^{\circ}\]