Challenging +1.8 This is a substantial Further Maths vectors question requiring multiple techniques: finding a plane normal via cross product, perpendicular distance from point to plane, parametric line equations, line-plane intersection, and angle between planes. While each individual step uses standard methods, the multi-part structure with 4-5 distinct calculations and the need to track multiple geometric objects makes this significantly harder than typical A-level questions, though not requiring exceptional insight.
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}}