Grade 12 Calculus and Vectors
Chapter 10: Equations of Planes in Three-Dimensions

In this chapter, we will be applying the knowledge and ideas from the previous chapter (as always) to planes. Developing vector and parametric equations for planes in three-dimensional space will be the focus in this section.

In order to describe a plane in vector and parametric form, we will need two parameters (whereas, lines only required one parameter). The table below exhibits some similarities between lines in two-dimension and planes in three-dimension. This is especially useful when you are required to sketch planes in three-dimensions.

There are four ways of creating unique planes:
1. there must exist three points that are not on a line (non-collinear) and on different planes
2. a line and a point (the point must not be on the line) will construct a plane
3. there must be two lines that are intersecting/crossing
4. two parallel and non-coincident lines will also help build a plane

Review Topics:
1. ideas behind vector equations and parametric equations
2. linear combinations of vectors
3. vector spanning

Comparison of Planes and Lines

Plane	Behaviour	Line 2-D	Behaviour
f(x) = x = 1	Plane crosses the x-axis at (1, 0, 0) and has an x-intercept of 1. Also, parallel to the yz-plane or x=0 plane.	f(x) = x = 1	Line crosses the x-axis at (1, 0) and has an x-intercept of 1. Also, parallel to the y-axis or x=0 axis.
f(x) = y = 2	Plane crosses the y-axis at (0, 2, 0) and has a y-intercept of 2. Also, parallel to the xz-plane or y=0 plane.	f(x) = y = 2	Line crosses the y-axis at (0, 2) and has a y-intercept of 2. Also, parallel to the x-axis or y=0 axis.
f(x) = z = 3	Plane crosses the z-axis at (0, 0, 3) and has a z-intercept of 3. Also, parallel to the xy-plane or z=0 plane.

10.1 Vector and Parametric Equations of Planes



10.2 Scalar or Cartesian Equations of Planes



10.3 Graphing Planes on 3-D Coordinate System