Edexcel A-Level Mathematics A2 Fp3 5 3 Equation of a Plane Questions

Practise Edexcel IAL FP3.5.3 by finding normals from directions, writing Cartesian or vector plane equations and applying point conditions.

Exam points

find a normal vector from two directions, then form a Cartesian plane equation
convert a plane between r·n=p and r=a+sb+tc when points and directions are given
use a plane equation with a given point or line condition to determine an unknown constant

[Maximum number: 3]

The plane $\Pi_1$ has vector equation

\mathbf r=\begin{smallmatrix}5\\3\\0\end{smallmatrix}+s\begin{smallmatrix}3\\0\\1\end{smallmatrix}+t\begin{smallmatrix}1\\-2\\2\end{smallmatrix}

where s and t are scalar parameters.

Determine a Cartesian equation for $\Pi_1$ .

\mathbf n=\begin{smallmatrix}3\\0\\1\end{smallmatrix}\times\begin{smallmatrix}1\\-2\\2\end{smallmatrix}=\begin{smallmatrix}2\\-5\\-6\end{smallmatrix}

\begin{smallmatrix}5\\3\\0\end{smallmatrix}\cdot\begin{smallmatrix}2\\-5\\-6\end{smallmatrix}=-5

\mathbf r\cdot\begin{smallmatrix}2\\-5\\-6\end{smallmatrix}=-5

2x-5y-6z+5=0

M1: calculates a normal vector from the two direction vectors.
M1: calculates the scalar product using a point in the plane.
A1: any correct Cartesian equation.

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Edexcel A-Level Mathematics A2 Fp3 5 3 Equation of a Plane Questions question 1