Geometric Algebra: Find unknown vector from two dot products

This resource is part of a series that I am preparing in support of Professor David Hestenes's goal of using Geometric Algebra(GA) to integrate high-school algebra,geometry, trigonometry,and physics into a coherent curriculum. Abstract To learn how to use the inner (“dot”) product to solve vector-algebra problems, we explore the following exercise, providing several user-manipulable constructions to aid in understanding the problem and interpreting solutions: Given two known vectors, a and b, and their dot products with an unknown vector x that is coplanar with them, determine x. We begin by reviewing what we’ve learned previously about vectors and vector algebra (including dot products). That review leads us to two strategies for solving the problem: (1) write x as a linear combination of a and b, then relate the givens (particularly and to inner products of a and b with that linear combination; and (2) use the geometric content of the givens to reformulate the problem as (in effect) one of finding the intersection of two lines. After solving the problem using dot products, we solve it using GA’s outer product, which we develop in an unconventional way that helps us derive useful relationships between inner products, outer products, and the unit bivector i. Using those relationships, we transform the outer-product solution into the solution obtained via Strategy 2. To show the equivalence of all of the solutions obtained in this document, we then transform the Strategy 1 solution into the Strategy 2 solution, noting the importance of being alert to the geometrical content of algebraic quantities that appear in the course of manipulating GA equations. Previous materials in this series: Dot Product: Geometric Significance and GeoGebra Coding ( Inner and Outer Products of Vectors Inscribed in a Circle ( Resources prepared by Professor Hestenes and Robert Rowley: LinkedIn group for collaborative development of pre-university materials on GA: Pre-University Geometric Algebra Wiki for the same purpose: