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 \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\). Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\). Find the position vector of the point \(Q\). Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

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 $A B C$. The point $P$ on the line-segment $O N$ is such that $O P = \frac { 3 } { 4 } O N$. The line $A P$ meets the plane $O B C$ at $Q$. Find a vector perpendicular to the plane $A B C$ and show that the length of $O N$ is $\frac { 4 } { \sqrt { } ( 21 ) }$.

Find the position vector of the point $Q$.

Show that the acute angle between the planes $A B C$ and $A B Q$ is $\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)$.

\hfill \mbox{\textit{CAIE FP1 2015 Q11 EITHER}}