Vector and Cartesian equations of planesAQA A-Level Further Maths: Revision notes

Section 1

The vector equation of a plane

A plane is fixed by a point $A$ and two non-parallel vectors $\mathbf b$ and $\mathbf c$ that lie in it: $\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c,$ where $\lambda$ and $\mu$ are independent parameters. Every pair of values gives a point on the plane. Through three points $A$ , $B$ , $C$ use $\mathbf b=\overrightarrow{AB}$ and $\mathbf c=\overrightarrow{AC}$ : for $A(1,2,0)$ , $B(3,2,1)$ , $C(2,5,2)$ the plane is $\mathbf r=(1,2,0)+\lambda(2,0,1)+\mu(1,3,2)$ .

Key termsvector equation of a planeparameter

Common mistake

Using position vectors of the other points as directions. The directions must be differences such as $\overrightarrow{AB}$ .

Exam tip

The two direction vectors must not be multiples of each other, otherwise you only describe a line.

Section 2

The normal vector and r·n = d

A normal vector $\mathbf n$ is perpendicular to every direction in the plane. If $\mathbf r$ and $\mathbf a$ are both in the plane, then $\mathbf r-\mathbf a$ lies in the plane, so $(\mathbf r-\mathbf a)\cdot\mathbf n=0$ , giving the scalar product form $\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n=d.$ To find $\mathbf n$ from directions $\mathbf b$ and $\mathbf c$ , solve $\mathbf n\cdot\mathbf b=0$ and $\mathbf n\cdot\mathbf c=0$ (two equations in three unknowns, so choose one value freely). Example: $\mathbf b=(1,2,0)$ and $\mathbf c=(0,3,1)$ give $n_1+2n_2=0$ and $3n_2+n_3=0$ . Take $n_2=-1$ : $\mathbf n=(2,-1,3)$ . With $\mathbf a=(1,0,2)$ , $d=2+0+6=8$ .

Key termsnormal vectorscalar product form

Common mistake

Using a direction vector instead of a point to find $d$ . Always use $d=\mathbf a\cdot\mathbf n$ with $\mathbf a$ a point on the plane.

Exam tip

Check $\mathbf n$ by taking its scalar product with both directions: both must be $0$ .

Section 3

The Cartesian equation

Writing $\mathbf r=(x,y,z)$ and $\mathbf n=(a,b,c)$ , the form $\mathbf r\cdot\mathbf n=d$ becomes $ax+by+cz=d.$ So $\mathbf r\cdot(2,-1,3)=8$ is $2x-y+3z=8$ . The coefficients of $x$ , $y$ and $z$ are the components of a normal vector. To test whether a point lies on the plane, substitute its coordinates: $(3,4,2)$ gives $6-4+6=8$ , so it lies on it. Parallel planes have the same normal (or multiples), so they differ only in $d$ : $3x-2y+z=12$ and $3x-2y+z=10$ are parallel.

Key termsCartesian equation of a plane

Common mistake

Reading the normal incorrectly from $x+y-2z=3$ : it is $(1,1,-2)$ , with the sign included.

Section 4

Converting between forms

Vector to Cartesian: find $\mathbf n$ from the directions, then $d=\mathbf a\cdot\mathbf n$ . Alternatively write $x=1+\lambda\ldots$ and eliminate $\lambda$ and $\mu$ . Cartesian to vector: read the normal, find three points by choosing values (for $3x-2y+z=12$ : $(4,0,0)$ , $(0,0,12)$ , $(0,-6,0)$ ), and use one as $\mathbf a$ with the differences as directions: $\mathbf r=(4,0,0)+\lambda(-1,0,3)+\mu(-2,-3,0)$ . Plane through a point with a given normal: use $\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n$ directly. For example a line perpendicular to a plane has the normal as its direction.

Key termsconversion

Exam tip

Find intercepts by setting two of $x,y,z$ to zero. They give three easy points on the plane.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Vector and Cartesian equations of planes

The plane $\Pi$ has vector equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}$ .
Determine whether the point $(3,4,2)$ lies on $\Pi$ .2 marks
The points $A(1,2,0)$ , $B(3,2,1)$ and $C(2,5,2)$ lie in a plane $\Pi$ .
Hence find a Cartesian equation of $\Pi$ .2 marks
The plane $\Pi$ has Cartesian equation $3x-2y+z=12$ .
Find a vector equation of $\Pi$ in the form $\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).