8 The points \(P , Q\) and \(R\) have coordinates \(( 0,2,3 ) , ( 2,0,1 )\) and \(( 1,3,0 )\) respectively.
The acute angle between the line segments \(P Q\) and \(P R\) is \(\theta\).
- Show that \(\sin \theta = \frac { 2 } { 11 } \sqrt { 22 }\).
The triangle \(P Q R\) lies in the plane \(\Pi\).
- Determine an equation for \(\Pi\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\), where \(a , b , c\) and \(d\) are integers.
The point \(S\) has coordinates \(( 5,3 , - 1 )\).
- By finding the shortest distance between \(S\) and the plane \(\Pi\), show that the volume of the tetrahedron \(P Q R S\) is \(\frac { 14 } { 3 }\).
[0pt]
[The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height]
The tetrahedron \(P Q R S\) is transformed to the tetrahedron \(\mathrm { P } ^ { \prime } \mathrm { Q } ^ { \prime } \mathrm { R } ^ { \prime } \mathrm { S } ^ { \prime }\) by a rotation about the \(y\)-axis.
The \(x\)-coordinate of \(S ^ { \prime }\) is \(2 \sqrt { 2 }\). - By using the matrix for a rotation by angle \(\theta\) about the \(y\)-axis, as given in the Formulae Booklet, determine in exact form the possible coordinates of \(R ^ { \prime }\).