The points \(A, B\) and \(C\) have position vectors \(2 \mathbf{i}-\mathbf{j}+\mathbf{k}, 3 \mathbf{i}+4 \mathbf{j}-\mathbf{k}\) and \(-\mathbf{i}+2 \mathbf{j}+4 \mathbf{k}\) respectively.
(i) Find the area of the triangle \(A B C\).

(ii) Find the perpendicular distance of the point \(A\) from the line \(B C\).

(iii) Find the cartesian equation of the plane through \(A, B\) and \(C\).