Equations of Lines and Planes in 3D - Wild Linear Algebra A - NJ Wildberger

Overview

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Explore equations of lines and planes in three-dimensional space in this comprehensive lecture on linear algebra. Begin with a review of lines in 2D, including their Cartesian and parametric equations, before delving into the novel concept of representing all 2D lines as a Möbius band. Progress to 3D geometry, examining both lines and planes using determinants rather than inner or cross products. Learn to find intersections, solve systems of equations, and convert between parametric and Cartesian forms for both lines and planes. Gain insights into special cases, pencils and stacks of lines, and unique methods for finding meets of planes rarely covered in traditional texts. This in-depth exploration provides a solid foundation for understanding the geometry of lines and planes in 3D space.

Syllabus

CONTENT SUMMARY: pg 1: @
pg 2: @
pg 3: @
pg 4: @
pg 5: @ special lines in the 2dimensional case; the x and y axes, and lines parallelel to the x and y axes;
pg 6: @ pencils and stacks;
pg 7: @ question: What does the space of all lines look like?; topologically gluing a line to every point on a circle;
pg 8: @ cylinder; Mobius band;
pg 9: @ lines and planes in 3D; planes; cartesian equation of a plane;
pg 10: @ solving a system of equations in 3D; matrix of determinants of minors;
pg 11: @ lines in 3D; two points, point and vector, intersection of 2 planes; parametric equation;
pg 12: @ line in cartesian and parametric form; cartesian form describes 2 planes that meet in a line;
pg 13: @ examples;
pg 14: @ meet of two planes; method found in very few linear algebra texts; a way of introducing parameters; THANKS to EmptySpaceEnterprise
Introduction
Meet of lines
Special lines
Pencils and stacks
What does the spaces of all lines look like?
Two possibilities
The analog of a line in 2D is a plane in 3D
Two planes generally meet in a line
Lines are determined by: two points, point + vector, two planes
Lines also have Cartesian equations
Meet of two planes
Parametric equation for a plane
Parametric to Cartesian plane
Cartesian to parametric plane