Show that the vectors `2 hat i- hat j+ hat k , hat i-3 hat j-5 hat k` and `3 hat i-4 hat j-4 hat k` form the vertices of a right angled triangle.

Explanation

To show that the vectors \( \vec{A} = 2\hat{i} - \hat{j} + \hat{k} \), \( \vec{B} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \vec{C} = 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle, we will follow these steps:

### Step 1: Find the vectors representing the sides of the triangle

1. **Calculate \( \vec{AB} \)**:
\[ \vec{AB} = \vec{B} - \vec{A} = (\hat{i} - 3\hat{j} - 5\hat{k}) - (2\hat{i} - \hat{j} + \hat{k}) \]
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