Step-by-step Solution:

1. Given Vectors: \[ \vec{A} = 4\hat{i} + 3\hat{j} + \hat{k} \] \[ \vec{B} = 2\hat{i} - \hat{j} + 2\hat{k} \] 2. Compute the Cross Product \(\vec{A} \times \vec{B}\): The cross product \(\vec{A} \times \vec{B}\) is perpendicular to both \(\vec{A}\) and \(\vec{B}\): \[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 3 & 1 \\ 2 & -1 & 2 \end{vmatrix} \] \[ \vec{A} \times \vec{B} = \hat{i}(3 \cdot 2 - 1 \cdot (-1)) - \hat{j}(4 \cdot 2 - 1 \cdot 2) + \hat{k}(4 \cdot (-1) - 3 \cdot 2) \] \[ \vec{A} \times \vec{B} = \hat{i}(6 + 1) - \hat{j}(8 - 2) + \hat{k}(-4 - 6) \] \[ \vec{A} \times \vec{B} = 7\hat{i} - 6\hat{j} - 10\hat{k} \] 3. Compute the Magnitude of \(\vec{A} \times \vec{B}\): \[ \|\vec{A} \times \vec{B}\| = \sqrt{7^2 + (-6)^2 + (-10)^2} = \sqrt{49 + 36 + 100} = \sqrt{185} \] 4. Find the Unit Vector \(\hat{N}\): The unit vector \(\hat{N}\) is the cross product divided by its magnitude: \[ \hat{N} = \frac{\vec{A} \times \vec{B}}{\|\vec{A} \times \vec{B}\|} = \frac{7\hat{i} - 6\hat{j} - 10\hat{k}}{\sqrt{185}} \] Therefore, the unit vector \(\hat{N}\) perpendicular to \(\vec{A}\) and \(\vec{B}\), forming a right-handed system, is: \[ \boxed{\frac{1}{\sqrt{185}} \left(7\hat{i} - 6\hat{j} - 10\hat{k}\right)} \]