Orthogonal Transformations Which Take Given Vectors into Given Vectors in R3

Abstract

For two given vectors in R3 we investigate the problem of identifying all orthogonal transformations (proper and improper rotations) which take one of them into the other. We extend our analysis to the problem of identifying all orthogonal transformations which take two vectors into two other vectors. Our analysis is based on the Euler-Rodrigues’ formula for orthogonal transformations in R3. Examples are giving for which there are infinitely many solutions, a unique solution, and no solution

Keywords: orthogonal transformation, propper rotation, improper rotation, Euler-Rodrigues formula, identification