21M.2.AHL.TZ1.6

pestleMathematicsAAHLPaper 221M· ahl-3-16-vector-productsource ↗

Consider the planes $Π_{1}$ and $Π_{2}$ with the following equations.

$Π_{1} : 3 x + 2 y + z = 6$

$Π_{2} : x - 2 y + z = 4$

Find a Cartesian equation of the plane $Π_{3}$ which is perpendicular to $Π_{1}$ and $Π_{2}$ and passes through the origin $(0, 0, 0)$ .

[3]

Find the coordinates of the point where $Π_{1}$ , $Π_{2}$ and $Π_{3}$ intersect.

[2]

Markscheme / solution

attempt to find a vector perpendicular to $Π_{1}$ and $Π_{2}$ using a cross product (M1)

$(\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}) \times (\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}) = (2 - (- 2)) i + (1 - 3) j + (- 6 - 2) k$

$= (\begin{matrix} 4 \\ - 2 \\ - 8 \end{matrix}) (= 2 (\begin{matrix} 2 \\ - 1 \\ - 4 \end{matrix}))$ (A1)

equation is $4 x - 2 y - 8 z = 0 (\Rightarrow 2 x - y - 4 z = 0)$ A1

[3 marks]

attempt to solve $3$ simultaneous equations in $3$ variables (M1)

$(\frac{41}{21}, - \frac{10}{21}, \frac{23}{21}) (= (1.95, - 0.476, 1.10))$ A1

[2 marks]

Examiners’ report

[N/A]