Vector Geometry: Lines and Planes in 3D Space
Defining Lines and Planes in 3D Space
Plane Definitions
Plane defined by two intersecting lines
Formed by the direction vectors of both lines and their intersection point.
Plane through a point, parallel to another plane
Take the normal vector of the parallel plane and the given point.
Plane through a point, perpendicular to a line
The given point and the direction vector of the line will be the normal vector of the plane.
Plane through a point (x₀, y₀, z₀), perpendicular to two planes
The normal vector n₃ is the cross product of the normal vectors of the two planes: n₃ = n₂ × n₁. The plane's equation is: n₃ₓ(x - x₀) + n₃ᵧ(y - y₀) + n₃𝓩(z - z₀) = 0.
Line Definitions
Line through a point, parallel to two planes
The direction