Show that the vectors
`hati-3hatj+2hatk,2hati-4hatj-hatk and 3hati+2hatj-hatk` and linearly independent.

Explanation

To show that the vectors \(\hat{i} - 3\hat{j} + 2\hat{k}\), \(2\hat{i} - 4\hat{j} - \hat{k}\), and \(3\hat{i} + 2\hat{j} - \hat{k}\) are linearly independent, we can form a matrix with these vectors as rows (or columns) and then calculate the determinant. If the determinant is not equal to zero, the vectors are linearly independent.

### Step-by-Step Solution:

1. **Write the vectors in component form:**
- Let \(\mathbf{v_1} = \hat{i} - 3\hat{j} + 2\hat{k} = (1, -3, 2)\)
- Let \(\mathbf{v_2} = 2\hat{i} - 4\hat{j} - \hat{k} = (2, -4, -1)\)
- Let \(\mathbf{v_3} = 3\hat{i} + 2\hat{j} - \hat{k} = (3, 2, -1)\)
...