Equations of a line and a plane in space
A line in space is fixed by a point and a direction vector, and a plane by a point and a normal vector. Write their equations and check points against them.
- UK: Year 13
- France: grade 11
- Germany: grade 12
- Iran: grade 12
- Syria: grade 12
- Russia: grade 11
Guide
This studio is for students who have met vectors in space. It is not in Afghanistan's Year 11 book; it follows Syria's Year 12 book, Iran's Year 12 geometry 3 and UK Further Maths. A line in space is written r = a + td: a is the position vector of a point on the line, d is a direction vector and t is any real number. Each t gives one point of the line.
A plane is fixed by one of its points and a normal vector n, perpendicular to the plane. A point P is on the plane if the vector from the fixed point to P is perpendicular to n, that is n·r = n·a. In Cartesian form: Ax + By + Cz = D, where ABC is n.
Common mistakes: starting from B instead of A, losing the minus signs of components, and adding the components instead of multiplying them in a scalar product.