Show that the four points with position vectors```4 hat i+8 hat j+12 hat k ,\ 2 hat i+4\ hat j+6 hat k ,3 hat i+5\ hat j+4 hat k\ a n d\ 5 hat i+8\ hat j+5 hat k`are coplanar.

Explanation

To show that the four points with position vectors \( \vec{A} = 4\hat{i} + 8\hat{j} + 12\hat{k} \), \( \vec{B} = 2\hat{i} + 4\hat{j} + 6\hat{k} \), \( \vec{C} = 3\hat{i} + 5\hat{j} + 4\hat{k} \), and \( \vec{D} = 5\hat{i} + 8\hat{j} + 5\hat{k} \) are coplanar, we can follow these steps:

### Step 1: Find the vectors \( \vec{AB} \), \( \vec{AC} \), and \( \vec{AD} \)

1. **Calculate \( \vec{AB} \)**:
\[ \vec{AB} = \vec{B} - \vec{A} = (2\hat{i} + 4\hat{j} + 6\hat{k}) - (4\hat{i} + 8\hat{j} + 12\hat{k}) = (-2\hat{i} - 4\hat{j} - 6\hat{k}) \]
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