Position of lines and planes in space
Point test for a line, normal vector by the cross product, the position of two lines and the intersection of a line and a plane.
- France: grade 11
- Germany: grade 12
- Syria: grade 12
Guide
A line in space has the parametric form g: X⃗ = A⃗ + λ · u⃗ with the support vector A⃗ and the direction vector u⃗. A point P lies on g if exactly one λ satisfies all three coordinates (point test).
A plane has the normal form E: n₁x₁ + n₂x₂ + n₃x₃ = d. The normal vector n⃗ for two direction vectors u⃗ and v⃗ comes from the cross product u⃗ × v⃗.
Two lines are identical, parallel (direction vectors are multiples), intersecting (exactly one point) or skew (not parallel and no intersection). A line meets a plane if you can put its term in and solve for λ.