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 direction cosines \(\left(\frac{12}{13}, -\frac{3}{13}, -\frac{4}{13}\right)\), \(\left(\frac{4}{13}, \frac{12}{13}, \frac{3}{13}\right)\), and \(\left(\frac{3}{13}, -\frac{4}{13}, \frac{12}{13}\right)\) are mutually perpendicular, we will use the property that two lines are perpendicular if the sum of the products of their direction cosines is zero.

### Step 1: Identify the direction cosines
Let:
- Line 1 (L1): \(L_1 = \frac{12}{13}, M_1 = -\frac{3}{13}, N_1 = -\frac{4}{13}\)
- Line 2 (L2): \(L_2 = \frac{4}{13}, M_2 = \frac{12}{13}, N_2 = \frac{3}{13}\)
- Line 3 (L3): \(L_3 = \frac{3}{13}, M_3 = -\frac{4}{13}, N_3 = \frac{12}{13}\)

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