The points \(A , B\) and \(C\) have coordinates ( \(3,2,2\) ), ( \(- 1,1,3\) ) and ( \(- 2,4,2\) ) respectively. The plane \(\Pi _ { 1 }\) contains the points \(A , B\) and \(C\)
Determine a Cartesian equation of \(\Pi _ { 1 }\)
Given that
point \(D\) has coordinates \(( - 1,1 , - 2 )\)
line \(l\) passes through \(D\) and is perpendicular to \(\Pi _ { 1 }\)
plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( 14 \mathbf { i } - \mathbf { j } - 17 \mathbf { k } ) = - 66\)
\(I\) meets \(\Pi _ { 2 }\) at the point \(E\)
show that \(D E = p \sqrt { 22 }\) where \(p\) is a rational number to be determined.
The point \(F\) has coordinates ( \(4,3 , q\) ) where \(q\) is a constant.
Given that \(A , B , C\) and \(F\) are the vertices of a tetrahedron of volume 12