Show that the three lines with direction cosines `(12)/(13),(-3)/(13),(-4)/(13),4/(13),(12)/(13),3/(13);3/(13),(-4)/(13),(12)/(13)`are mutually perpendicular.

Explanation

To show that the three lines with the given direction cosines are mutually perpendicular, we need to verify that the dot product of each pair of direction vectors is zero.

### Step-by-Step Solution:

1. **Identify the Direction Cosines:**
- Let the first direction cosine vector \( \mathbf{a} = \left( \frac{12}{13}, -\frac{3}{13}, -\frac{4}{13} \right) \)
- Let the second direction cosine vector \( \mathbf{b} = \left( \frac{4}{13}, \frac{12}{13}, \frac{3}{13} \right) \)
- Let the third direction cosine vector \( \mathbf{c} = \left( \frac{3}{13}, -\frac{4}{13}, \frac{12}{13} \right) \)
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