A tetrahedron has vertices \(A ( 1,2,1 ) , B ( 0,1,0 ) , C ( 2,1,3 )\) and \(D ( 10,5,5 )\).
Find
a Cartesian equation of the plane \(A B C\).
the volume of the tetrahedron \(A B C D\).
The plane \(\Pi\) has equation \(2 x - 3 y + 3 = 0\)
The point \(E\) lies on the line \(A C\) and the point \(F\) lies on the line \(A D\).
Given that \(\Pi\) contains the point \(B\), the point \(E\) and the point \(F\),
find the value of \(k\) such that \(\overrightarrow { A E } = k \overrightarrow { A C }\).
Given that \(\overrightarrow { A F } = \frac { 1 } { 9 } \overrightarrow { A D }\)
show that the volume of the tetrahedron \(A B C D\) is 45 times the volume of the tetrahedron \(A B E F\).