Key Idea: A plane needs one point and a normal n, a vector sticking straight out of it. This topic writes planes in two forms and builds them from points or lines. It is part of the long 3D vectors question.
Equation of a plane
- Scalar-product form: r·n = a·n.
- Cartesian form: ax + by + cz = d, where (a, b, c) is the normal.
- Is a point on the plane? Substitute it and check both sides.
- Any multiple of n is still a normal.
| You are given | Normal n |
|---|---|
| Three points A, B, C | AB × AC |
| A point P and a line with direction d | d × AP |
| r = a + λu + μv | u × v |
Every plane question starts with finding a normal.
Plane through C(−1, 4, 2) with normal (1, 3, −2)? x + 3y − 2z = 7.
Normal of 2x − y + 3z = 5? (2, −1, 3).
You simplify n = (4, −2, 4) to (2, −1, 2). What happens to k = 6? It also halves, to 3.
How do you find d once you have n? Substitute any known point into ax + by + cz.