Standard +0.8 This is a substantial multi-part Further Maths question requiring: finding a plane equation via cross product, computing triangle area using vector methods, calculating tetrahedron volume (either via scalar triple product or base×height), and finding the angle between two planes. While each technique is standard for FM students, the question demands accurate execution across multiple steps with 3D vectors, placing it moderately above average difficulty.
The points \(A , B , C\) and \(D\) have coordinates as follows:
$$A ( 2,1 , - 2 ) , \quad B ( 4,1 , - 1 ) , \quad C ( 3 , - 2 , - 1 ) \quad \text { and } \quad D ( 3,6,2 ) .$$
The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(C\). Find a cartesian equation of \(\Pi _ { 1 }\).
Find the area of triangle \(A B C\) and hence, or otherwise, find the volume of the tetrahedron \(A B C D\). [0pt]
[The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.]
The plane \(\Pi _ { 2 }\) passes through the points \(A , B\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
The points $A , B , C$ and $D$ have coordinates as follows:
$$A ( 2,1 , - 2 ) , \quad B ( 4,1 , - 1 ) , \quad C ( 3 , - 2 , - 1 ) \quad \text { and } \quad D ( 3,6,2 ) .$$
The plane $\Pi _ { 1 }$ passes through the points $A , B$ and $C$. Find a cartesian equation of $\Pi _ { 1 }$.
Find the area of triangle $A B C$ and hence, or otherwise, find the volume of the tetrahedron $A B C D$.\\[0pt]
[The volume of a tetrahedron is $\frac { 1 } { 3 } \times$ area of base × perpendicular height.]\\
The plane $\Pi _ { 2 }$ passes through the points $A , B$ and $D$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\hfill \mbox{\textit{CAIE FP1 2013 Q11 OR}}