Step-by-step Solution:

The correct option is \[ \mathbf{D)} \quad \hat{i} - \hat{j} - 5\hat{k} \] \[ \mathbf{r} = \mathbf{f}_1 - \mathbf{f}_2 = 6\hat{i} + \hat{j} + \hat{k} \] \[ \mathbf{F} = 5\hat{i} + \hat{k} \] Moment of the couple is given by \[ \mathbf{r} \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 1 & 1 \\ 5 & 0 & 1 \end{vmatrix} \] Expanding the determinant: \[ = \hat{i} \begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 6 & 1 \\ 5 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 6 & 1 \\ 5 & 0 \end{vmatrix} \] \[ = \hat{i} (1 \cdot 1 - 1 \cdot 0) - \hat{j} (6 \cdot 1 - 1 \cdot 5) + \hat{k} (6 \cdot 0 - 1 \cdot 5) \] \[ = \hat{i} (1) - \hat{j} (6 - 5) + \hat{k} (0 - 5) \] \[ = \hat{i} - \hat{j} - 5\hat{k} \]