3.17 Equations of a plane (IB Maths: Analysis and Approaches HL): Revision notes

Section 1

What fixes a plane?

A plane is fixed by one point on it together with either

two non-parallel vectors that lie in the plane, or
one normal vector, perpendicular to every line in the plane.

These give the three forms of the equation of a plane in the formula booklet. Three points that are not in a straight line also fix a plane: use one as the point and the two vectors joining it to the others as the directions.

Key termsnormal vectorcoplanar

Exam tip

Any non-zero multiple of a normal is also a normal. Scale it to remove fractions and common factors.

Section 2

The vector (parametric) form

$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}, \quad \lambda, \mu\in\mathbb{R},$ where $\mathbf{a}$ is the position vector of a point on the plane and $\mathbf{b}$ , $\mathbf{c}$ are non-parallel vectors in the plane. Each pair of values $(\lambda, \mu)$ gives one point.

For P $(1,0,2)$ , Q $(3,1,0)$ , R $(0,2,1)$ : $\mathbf{r} = (\mathbf{i} + 2\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + \mu(-\mathbf{i} + 2\mathbf{j} - \mathbf{k})$ , using $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ .

To find the parameters for a given point, equate components and solve two of the equations; the third must then be satisfied.

Key termsvector equation of a planeparameters

Common mistake

Using position vectors of points as direction vectors. The directions must be vectors between points of the plane, such as $\overrightarrow{PQ}$ .

Common mistake

Choosing two parallel direction vectors. They only describe a line, not a plane.

Section 3

The scalar product form and the Cartesian form

Every point R on the plane satisfies $\overrightarrow{AR}\cdot\mathbf{n} = 0$ , which rearranges to $\mathbf{r}\cdot\mathbf{n} = \mathbf{a}\cdot\mathbf{n}.$ Writing $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ and $\mathbf{n} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ gives the Cartesian equation $ax + by + cz = d, \quad d = \mathbf{a}\cdot\mathbf{n}.$ The coefficients of $x, y, z$ are a normal vector.

Example: through $(1, 2, -1)$ with $\mathbf{n} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ : $d = 2 - 2 - 3 = -3$ , so $2x - y + 3z = -3$ .

Key termsCartesian equation of a plane

Exam tip

A point lies on the plane exactly when its coordinates satisfy the Cartesian equation. This is the quickest check.

Common mistake

Setting $d = 0$ automatically. The plane only passes through the origin if $\mathbf{a}\cdot\mathbf{n} = 0$ .

Section 4

Converting between the forms

Vector to Cartesian: find $\mathbf{n} = \mathbf{b}\times\mathbf{c}$ , then $d = \mathbf{a}\cdot\mathbf{n}$ . For $\mathbf{r} = (1,1,0) + \lambda(1,0,2) + \mu(0,1,-1)$ : $\mathbf{n} = (-2, 1, 1)$ and $d = -1$ , giving $2x - y - z = 1$ .

Three points to Cartesian: form two vectors between the points, take their vector product, then substitute one point.

Cartesian to vector: find any point on the plane (set two variables to 0) and two non-parallel vectors perpendicular to $\mathbf{n}$ (check their scalar product with $\mathbf{n}$ is 0).

Exam tip

After finding a Cartesian equation from three points, substitute the other two points as a check. It takes seconds and catches sign slips in the vector product.

Section 5

Parallel planes and planes in context

Two planes are parallel when their normals are parallel. They are the same plane only if the equations are multiples of each other, constants included: $2x - y - z = 1$ and $4x - 2y - 2z = 7$ are parallel but distinct, because the second is $2x - y - z = 3.5$ .

In modelling questions a plane may represent a roof, a panel or a slope. If you know $x$ and $y$ for a point on the surface, substitute into the Cartesian equation to find its height $z$ . For the panel $9x - 6y + 8z = 26$ , the point above $(3, 2, 0)$ has $z = \frac{11}{8}$ .

Key termsparallel planes

Must know

Vector form: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$ , $\mathbf{b}$ and $\mathbf{c}$ non-parallel and in the plane.
Scalar product form: $\mathbf{r}\cdot\mathbf{n} = \mathbf{a}\cdot\mathbf{n}$ .
Cartesian form: $ax + by + cz = d$ , with normal $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ .
A normal from two vectors in the plane: $\mathbf{n} = \mathbf{b}\times\mathbf{c}$ .
Parallel planes have parallel normals; check the constants to see if they coincide.

That's the notes covered.

Carry on to the next subtopic.