# Equations of Planes (IB Mathematics AA HL) | Maths with Lemon

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### Equations of Planes

IB Mathematics: Analysis and Approaches Higher Level · Lesson V.6 · AHL 3.17

Learn to describe planes in three dimensions using vector, scalar-product and Cartesian equations, and convert between their forms.

Lesson V.6 worksheet (PDF) · Vectors overview


#### 1. What Defines a Plane?


##### Key ideas

- Three non-collinear points define a unique plane.
- One point and two non-parallel direction vectors also define a unique plane.
- A point and a non-zero normal vector determine a plane.
- Two points alone lie on infinitely many planes.

##### Geometry in 3D

The parameters \(\lambda\) and \(\mu\) vary independently, generating every point on the plane.

Why must \(\mathbf b\) and \(\mathbf c\) be non-parallel? If they were parallel, the equation would describe a line rather than a plane.


#### 2. Vector Equation of a Plane


##### Steps

- Take the position vector of a known point.
- Choose two non-parallel vectors lying in the plane.
- Introduce independent parameters.
- To test a point, solve all three coordinate equations using the same two parameters.

##### Worksheet example

For \(B(3,-2,5)\), the \(y\)-coordinate requires \(\lambda=-3\). The \(x\)-coordinate then requires \(\mu=-3/2\), but the resulting \(z\)-coordinate is \(-25/2\), not \(5\). Thus \(B\) is not on the plane.


#### 3. Scalar-Product Form


##### Normal vectors

- A normal vector is perpendicular to every direction vector in the plane.
- The displacement between any two points of the plane is perpendicular to the normal.
- Planes with parallel normal vectors are parallel or identical.

##### Point–normal equation


#### 4. Cartesian Equation of a Plane


##### What to remember

- The coefficients of \(x,y,z\) form a normal vector.
- Substitute any point on the plane to determine the constant.
- A Cartesian equation describes an entire plane, not just one point.

##### Standard form

Verify by substituting \(A(4,8,-1)\).


#### 5. Vector to Cartesian: Cross Product


##### Conversion method

- Read the two direction vectors from the vector equation.
- Find their cross product to obtain a normal.
- Use a point from the vector equation to calculate \(d\).
- Write \(ax+by+cz=d\) and simplify.

##### Worked example from the worksheet


#### 6. Plane Through Three Points


##### Construction

- Select one point \(A\) as the starting position.
- Form the direction vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).
- Write the vector equation.
- Cross the direction vectors to find a normal and then a Cartesian equation.

##### Worksheet example


#### 7. Plane Containing Intersecting Lines


##### Procedure

- Solve the two line equations to locate their intersection.
- Take the two line direction vectors as plane directions.
- Use the intersection as the plane’s fixed point.
- Cross the direction vectors to find the normal.

##### Key formulas

Here \(\mathbf r_0\) is the common point of the two intersecting lines. If the directions are parallel, a different construction is required.


#### 8. Distance from a Point to a Plane


##### Shortest distance

- The shortest segment is perpendicular to the plane.
- Use the normal-vector coefficients in the denominator.
- Substitute the point into the left-hand side and subtract the plane constant.

##### Formula and worksheet example


#### 9. Interactive Plane Explorer


##### Explore a plane in 3D

- Adjust the coefficients in \(ax+by+cz=d\).
- Rotate the diagram to inspect the orientation.
- The arrow shows the normal vector.
- The rendered grid lies on the plane, with perspective projected onto the screen.

##### Visualise the normal and plane


#### 10. HL Practice and Revision


##### From Lesson V.6

- Write vector equations and test whether points belong to planes.
- Find scalar-product and Cartesian forms from a normal and point.
- Convert vector equations to Cartesian equations using cross products.
- Determine planes from three non-collinear points.
- Find a plane containing two intersecting lines or a line and an external point.
- Calculate the shortest distance from the origin to a plane.

##### Common mistakes

- Using parallel direction vectors to define a plane.
- Confusing a direction vector with a normal vector.
- Taking the dot product instead of the cross product to find a normal.
- Forgetting to substitute a known point to calculate \(d\).
- Mixing the two parameters when checking whether a point lies on a plane.
Open the complete worksheet

Worksheet video: visualising scalar and vector products


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