Contents
The Five Lessons
Work through them in order. Each one opens with what you should be able to do by the end, develops the ideas with diagrams you can rotate, closes with three or four worked examples graded from routine to demanding, and finishes with the mistakes that most often cost a mark.
Lesson 01
Equations of Lines in R² and R³
Vector, parametric, symmetric and Cartesian forms; converting between
them; why R³ needs more than one equation; and what to do when a direction component is
zero. Interactive: a live form converter.
Lesson 02
Equations of Planes
A point and two directions, or a point and a normal — and the cross
product that converts between them. Five ways to define a plane, and how to run a Cartesian
equation backwards. Interactive: a determinant expander and a plane builder.
Lesson 03
Plotting and Sketching in R³
The right-handed frame, the projection box, the intercept triangle,
intercept form, traces, and what a missing variable tells you. Interactive: a point
plotter and an intercept-triangle builder, both rotatable.
Lesson 04
Intersections and Consistent Systems
Two lines — including the skew case that only exists in three-space
— a line and a plane, and two planes; and what consistent really means.
Interactive: a case explorer and a two-plane solver.
Lesson 05
Distances from Points to Lines and Planes
Three formulas and one idea: keep only the perpendicular part. Includes
the distance between two parallel planes, and the trap that formula sets.
Interactive: a distance lab that works every substitution line by line.