Step-by-Step Solution

1. Check for Parallelism: Two vectors are parallel if one is a scalar multiple of the other. Here, \(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\). There is no scalar \(k\) such that \(\vec{a} = k\vec{b}\). So, \(\vec{a}\) is not parallel to \(\vec{b}\).

2. Check for Perpendicularity: Two vectors are perpendicular if their dot product is zero. \(\vec{a} \cdot \vec{b} = (3)(1) + (2)(-1) + (-1)(1) = 3 - 2 - 1 = 0\). Therefore, \(\vec{a} \perp \vec{b}\).

3. Calculate Magnitudes: \(|\vec{a}| = \sqrt{(3)^2 + (2)^2 + (-1)^2} = \sqrt{9 + 4 + 1} = \sqrt{14}\) \(|\vec{b}| = \sqrt{(1)^2 + (-1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}\)

4. Compare Magnitudes: Since \(\sqrt{14} \neq \sqrt{3}\), \(|\vec{a}| \neq |\vec{b}|\). Also, since \(\sqrt{3} < \sqrt{14}\), \(|\vec{b}| < |\vec{a}|\).

5. Conclusion: From the above steps, we find that \(\vec{a} \perp \vec{b}\) is the correct statement.