The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line segment \(ON\) is such that \(OP=\frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\).

Find a vector perpendicular to the plane \(ABC\), and show that the length of \(ON\) is \(\frac{4}{\sqrt{21}}\).

Find the position vector of \(Q\).

Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{2}{3}\right)\).