A normal vector to a plane is any non-zero vector that is perpendicular to every direction lying within that plane. It determines the plane's orientation in three-dimensional space.
In IB Mathematics: Analysis and Approaches HL, this is part of AHL 3.17: Vector equations of a plane. If a plane contains two non-parallel direction vectors and , a normal vector satisfies
Therefore, one normal vector can be calculated using the cross product:
For example, if and , then . Check that and , so is perpendicular to both in-plane directions. Reversing the cross-product order gives , which is equally valid. The direction vectors must be non-parallel; otherwise their cross product is the zero vector, which cannot be a normal vector.
A plane can be represented in several equivalent forms:
| Form of plane | How to identify the normal vector |
|---|---|
| Cartesian: | |
| Scalar product: | The vector is stated directly. |
| Vector: | Calculate . |
For example, consider the plane
The coefficients of , , and give a normal vector:
Any non-zero scalar multiple, such as , is also normal to the same plane. A plane therefore has infinitely many normal vectors, all parallel or antiparallel to one another.
A common misconception is that a normal vector must start at a particular point on the plane. It does not: a vector represents magnitude and direction, while a separate position vector identifies a point through which the plane passes.
For a find question, show the cross product or extract the coefficients from the Cartesian equation before stating the vector. This AHL concept may be tested analytically on Paper 1 or with technology-assisted calculations on Papers 2 and 3; always verify perpendicularity using a dot product when appropriate.
