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]