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Geometry with Vectors - Wild Linear Algebra A 2 - NJ Wildberger

Insights into Mathematics via YouTube

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Explore fundamental concepts of vector geometry in this 44-minute lecture from the "Wild Linear Algebra" course by N J Wildberger at UNSW. Learn basic vector constructions, arithmetic operations, and affine combinations. Discover how to translate, add, subtract, and bisect vectors. Examine important theorems about parallelograms and triangle medians. Apply vector concepts to solve geometric problems, including proofs for the bisection of parallelogram diagonals and the concurrency of triangle medians. Gain insights into linear independence and parametric line descriptions using affine combinations. Practice with exercises on Varignon's theorem and other vector applications to enhance understanding of this geometric approach to linear algebra.

Syllabus

CONTENT SUMMARY: pg 1: @ how to approach the course; geometry with vectors
pg 2: @ constructions; translate a vector; create equally spaced points; add two vectors
pg 3: @ constructions; bisect a segment; trisect a segment; subtract vectors
pg 4: @ Vector Arithmetic
pg 5: Affine Combinations;
pg 6: general affine combination of vectors a and b; a way of describing the line parametrically; coefficients of the vectors add up to 1;
pg 7: The zero vector; Linear independence;
pg 8: Theorem: The diagonals of a parallelogram bisect each other; proof;
pg 9: Theorem: The medians of a triangle are concurrent, meeting at a point G which divides each in the proportion 2:1 ; proof
pg 10: Theorem concerning the ratio of 2 parallel vectors a and b contained by 2 vectors c and d radiating from the same point; exercise 2.1 Varignon's theorem
pg 11: exercise 2.2; exercise 2.3 THANKS to EmptySpaceEnterprise
Introduction
Constructions
Vector Arithmetic
Affine combinations
A special vector
Diagonals of a parallelogram