17M.2.AHL.TZ1.H_7

pestleMathematicsAAHLPaper 217M· ahl-3-17-vector-equations-of-a-planesource ↗

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 $x + 2y - z - 6 = 0$ .

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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

row reduction M1

equation of plane Π is $x - y - z = 3$ or equivalent A1

[6 marks]

Examiners’ report

[N/A]