Show that the vectors `a =3hati - 2hatj+hatk, b=hati - 3hatj+5hatk` and `c=2hati+hatj-4hatk` form a right angled triangle.

Explanation

To show that the vectors \( \mathbf{a} = 3\hat{i} - 2\hat{j} + \hat{k} \), \( \mathbf{b} = \hat{i} - 3\hat{j} + 5\hat{k} \), and \( \mathbf{c} = 2\hat{i} + \hat{j} - 4\hat{k} \) form a right-angled triangle, we can use the property that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

### Step 1: Calculate the vectors \( \mathbf{b} + \mathbf{c} \)

We start by adding vectors \( \mathbf{b} \) and \( \mathbf{c} \):

\[
\mathbf{b} + \mathbf{c} = (\hat{i} - 3\hat{j} + 5\hat{k}) + (2\hat{i} + \hat{j} - 4\hat{k})
...