Find the Cartesian equation of plane Π containing the points \({\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)\) and \({\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)\) and perpendicular to the plane \(x + 2y - z - 6 = 0\).

METHOD 1

\(\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 3} \\ 0 \end{array}} \right)\)     (A1)

\(\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 3} \\ 0 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)\)     M1A1

\( = \left( {\begin{array}{*{20}{c}} 3 \\ { - 3} \\ { - 3} \end{array}} \right)\)     A1

\(x - y - z = k\)     M1

\(k = 3\) equation of plane Π is \(x - y - z = 3\) or equivalent     A1

METHOD 2

let plane Π be \(ax + by + cz = d\)

attempt to form one or more simultaneous equations:     M1

\(a + 2b - c = 0\)     (1)     A1

\(6a + 2b + c = d\)     (2)

\(3a - b + c = d\)     (3)     A1

Note:     Award second A1 for equations (2) and (3).

attempt to solve     M1

EITHER

using GDC gives \(a = \frac{d}{3},{\text{ }}b = - \frac{d}{3},{\text{ }}c = - \frac{d}{3}\)     (A1)

equation of plane Π is \(x - y - z = 3\) or equivalent     A1

OR

row reduction     M1

equation of plane Π is \(x - y - z = 3\) or equivalent     A1

[6 marks]