CAIE A-Level Further Math AS 1.6.1 Plane Equations Questions

Practise finding Cartesian equations of planes from points, vectors and lines.

Exam points

find Cartesian equations of planes
use vectors and points to determine plane equations
solve plane equation problems in context

[Maximum number: 5]

The points A, B, C have position vectors

\mathbf{i}+\mathbf{j}, \quad-\mathbf{i}+2 \mathbf{j}+4 \mathbf{k}, \quad-2 \mathbf{i}+\mathbf{j}+3 \mathbf{k},

respectively, relative to the origin O.

Find the equation of the plane ABC, giving your answer in the form a x+b y+c z=d.

$\overrightarrow{A B}=-2 \mathbf{i}+\mathbf{j}+4 \mathbf{k} \quad \overrightarrow{A C}=-3 \mathbf{i}+3 \mathbf{k} \quad \overrightarrow{B C}=-\mathbf{i}-\mathbf{j}-\mathbf{k}$
Finds direction vectors of two lines in the plane.

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} -2 1 -3 0 3\end{array}\right|=\left(\begin{array}{c}3 -6 3\end{array}\right) \sim\left(\begin{array}{c}1 -2 1\end{array}\right)$

M1 A1FT

Finds normal to the plane ABC.

$1(1)-2(1)+1(0)=-1 \Rightarrow x-2 y+z=-1$
Substitutes point. CAO

5

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CAIE A-Level Further Math AS 1.6.1 Plane Equations Questions question 1