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 \( \mathbf{A} = 2\hat{i} - \hat{j} + \hat{k} \), \( \mathbf{B} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \mathbf{C} = 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle, we will calculate the lengths of the sides of the triangle formed by these vectors and then check if the Pythagorean theorem holds.

### Step 1: Identify the vertices
Let:
- Point A correspond to vector \( \mathbf{A} = (2, -1, 1) \)
- Point B correspond to vector \( \mathbf{B} = (1, -3, -5) \)
- Point C correspond to vector \( \mathbf{C} = (3, -4, -4) \)

...