CAIE A-Level Further Math 1.6.1 Plane equations

Practise forming and converting plane equations from points, direction vectors and normals, usually giving answers as ax+by+cz=d.

Exam points

use two direction vectors in a plane to find a normal by vector product
substitute a point on the plane to determine d in ax+by+cz=d
convert between scalar and vector plane forms when solving geometry problems

[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.

$\overrightarrow{A B}=-2 \mathbf{i}+\mathbf{j}+4 \mathbf{k} \quad \overrightarrow{A C}=-3 \mathbf{i}+3 \mathbf{k} \quad \overrightarrow{B C}=-\mathbf{i}-\mathbf{j}-\mathbf{k}$

B1

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.

$1(1)-2(1)+1(0)=-1 \Rightarrow x-2 y+z=-1$

M1 A1

Substitutes point. CAO

5

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1.6.1—Plane equations question 1