## 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
**

## Grade 12 Calculus and Vectors

Chapter 1: Introductin to* Calculus *and Vectors

Chapter 2: Derivatives

Chapter 3: Applications of Derivatives

Chapter 4: Curve Sketching

Chapter 5: Derivatives of Trigonometric Functions

Chapter 6: Derivatives of Exponential Functions

Chapter 7: Introduction to * Vectors *

Chapter 8: Vector Operations and Applications

Chapter 9: Equations of Lines

Chapter 10: Equations of Planes

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

Chapter 11: Points, Lines and Planes

Videos for each topic contain a playlist, allowing you to choose the instructor and learning style to your liking. You can choose different videos in a topic by placing the mouse over the video and selecting a different video from the playlist. Videos of solutions to multiple example problems are also available to illustrate to students how each tools and techniques should be used.

Please refer to high school math prerequisite tree to see the recommended courses and materials one should be familiar with prior to taking up Calculus and Vectors.

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