ABSTRACT
In section 1 we introduced the vector space V of all equivalence classes of arrows in order to be able to formulate facts about the mutual location of points in space. This allowed us to express notions like parallelity or division ratio in terms of vector equations. If we further specified an origin O we could even identify points with elements of V by identifying each point P with its position vector p = O P → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq426.tif"/> ; however, this identification depended on the choice of O which is arbitrary. For example, if we choose O as the origin in the picture below then we identify P with v and Q with w; on the other hand, if O′ is chosen as the origin then P and Q are identified with v′ and w′, respectively. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/page95_1.tif"/>
