Equations of a plane in 3D space using vector formĀ ār=a+Ī»b+Ī¼c,Ā scalar product formĀ ārā n=aā n,Ā
A plane in 3D space can be described by a vector equation involving a fixed point and two direction vectors lying in the plane. Planes are often denoted byĀ āĪ āĀ (capital pi).
If the position vector of the fixed point isĀ āa,Ā and two non-parallel direction vectors in the plane areĀ ābāĀ andĀ āc,Ā then the plane is represented by:
Here,Ā āĪ»āĀ andĀ āĪ¼āĀ are parameters that can take any real values, allowingĀ ārāĀ to move freely across the entire surface of the plane.
The scalar product form of a plane uses a vector perpendicular ("normal") to the plane and one known point in the plane. If a point in the plane has position vectorĀ āaāĀ andĀ ānāĀ is a normal vector, then any other pointĀ ārāĀ lies in the plane if:
This equation expresses the idea that the vector from the known point to any other point in the plane is always perpendicular toĀ ān.
The Cartesian equation of a plane with a normal vectorĀ ānāĀ and containing a point with position vectorĀ āaāĀ is
whereĀ ān=āāān1ān2ān3āāā āā,d=aā n.
Nice work completing Equations of a plane, here's a quick recap of what we covered:
Exercises checked off