Equation of a Plane and Angles
A plane with unit normal n at distance d from the origin satisfies r.n = d, with Cartesian form ax + by + cz = k where (a, b, c) is the normal.
What is Equation of a Plane and Angles?
A plane with unit normal n at distance d from the origin satisfies r.n = d, with Cartesian form ax + by + cz = k where (a, b, c) is the normal.
Key formula / rule: The coefficients in the Cartesian form are the components of the normal.
Key points
- Compute the distance from a point to a plane in space.
- Analyse the relative position of a line and a plane.
Common exam trap
Taking the angle between a line and a plane as the angle with the normal.
Definitions
- Term
Equation of a Plane and Angles
- Meaning
A plane with unit normal n at distance d from the origin satisfies r.n = d, with Cartesian form ax + by + cz = k where (a, b, c) is the normal.
- Term
Equation of a Plane and Angles — explanation
- Meaning
Once the normal is identified, distances and angles reduce to dot products. The angle between a line and a plane is the complement of the angle between the line and the normal.
Learning objectives
Compute the distance from a point to a plane in space.
Analyse the relative position of a line and a plane.
Formulae
- Key point
The coefficients in the Cartesian form are the components of the normal.
- Key point
Distance from a point to a plane uses the normalised normal.
- Key point
A line lies in a plane when its direction is perpendicular to the normal and one point satisfies the plane.
Prerequisites
MAT-U4-TD-T1-S1-C1
Common mistakes
Taking the angle between a line and a plane as the angle with the normal.
Forgetting to normalise the normal in the distance formula.
Keywords
Equation
Plane
Angles
Practice preview
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