Show that the points `A,B` and C having position vectors `(3hati - 4hatj - 4hatk), (2hati - hatj + hatk)`and `(hati - 3hatj - 5hatk)` respectively, from the vertices of a right-angled triangle.

Explanation

To show that the points A, B, and C with position vectors \( \vec{A} = 3\hat{i} - 4\hat{j} - 4\hat{k} \), \( \vec{B} = 2\hat{i} - \hat{j} + \hat{k} \), and \( \vec{C} = \hat{i} - 3\hat{j} - 5\hat{k} \) form a right-angled triangle, we need to calculate the lengths of the sides of the triangle formed by these points and verify the Pythagorean theorem.

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

1. **Calculate vector AB:**
\[ \vec{AB} = \vec{B} - \vec{A} = (2\hat{i} - \hat{j} + \hat{k}) - (3\hat{i} - 4\hat{j} - 4\hat{k}) \]
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