Lines and Planes in 3D
Parametric and normal equations.
Key idea
Parametric and normal equations.
In depth
A line is given by a point and a direction vector: r = p + td. A plane is given by a normal vector n and a point: n · (r − p) = 0, or ax + by + cz = d. Two planes meet in a line unless parallel.
Worked example
The plane with normal (1, 2, 3) through the origin is x + 2y + 3z = 0.
Formula
Real-world uses
- Modelling flight paths and the planes of walls and roofs in CAD and architecture
Common mistake
Using the direction vector in the plane equation in place of the normal.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Lines and Planes in 3D?
Parametric and normal equations. It is a topic at the expert level in Geometry.
What is the formula for Lines and Planes in 3D?
The key formula is ax+by+cz = d.
Where is Lines and Planes in 3D used in real life?
Real-world examples: Modelling flight paths and the planes of walls and roofs in CAD and architecture.
What should I learn before Lines and Planes in 3D?
Make sure you are comfortable with 3D Coordinates, the previous topic in Geometry.
What is a common mistake with Lines and Planes in 3D?
Using the direction vector in the plane equation in place of the normal.
How can I practise Lines and Planes in 3D?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Lines and Planes in 3D?
Move on to Geometric Probability next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.