Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III
Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis

Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Planes

Overview

A plane in space is the simplest of 3D surfaces, infinitely large and flat, easy to describe based on where it intersects the three coordinate axes. Our analysis will include the intersection of a plane and a line, or that of two non-parallel planes.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 13.6.1 Understand the definition of a plane in R3 algebraically and geometrically
• 13.6.2 Find and interpret equation of the plane through point P0 with normal vector vecn
• 13.6.3 Find and interpret equation of the plane through three points
• 13.6.4 Analyze a plane's properties (normal vectors, axis intersections, intersections with the xy, xz, and yz planes)
• 13.6.5 Determine if two planes are parallel or orthogonal
• 13.6.6 Find coordinates of the point where a line and plane intersect
• 13.6.7 Find equation of the line where two planes intersect
• 13.6.8 Find distance between a point to a line
• 13.6.9 (Optional) Think about how you would find the distance from a point to a plane and from a plane to a plane

Terminology

Define: plane in R3, planes spanned by axes (ie. xy-plane, xz-plane, yz-plane)

Formulae to have in your notes: equation of a plane (scalar and vector forms) and distance between a point and a line

 Equation of a Plane through point P_o (x_o,y_o,z_o) normal to vector vecn=<> Scalar form (a.k.a. Component form) a(x-x_o)+b(y-y_o)+c(z-z_o)=0 -or- ax+by+cz=d where d=ax_o + by_o + cz_o Vector form vecn • vec(P_o P) = 0

 Distance Between a Point and a Line Distance d between the point Q and the line vecr = vec(r_o) + t vecv is d = || vecv xx vec(PQ) || / || vecv || where P is any point on the line and vecv is a vector parallel to the line.

Supplemental Resources (optional)

Paul's OL Notes - Calc III: Equations of Planes

Patrick JMT Just Math Tutorials:
Finding the Scalar Equation of a Plane
Finding the Point where a Line Intersects a Plane

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Vectors in Space". There are lots of related titles in the second half of that list starting with The Equation of a Plane in 3D Using Vectors.