In the "intro to differential forms" thread by lethe, Super Mentor Tom defines a vector as something that transforms under rotation (multiplication by an orthogonal matrix) and parity (reflection through a mirror) in a certain way. I'm currently reading "Introduction to Vector and Tensor Analysis" by Robert Wrede, which uses the transformations of rotation and translation instead. So I have two questions: 1). Why explicitly define how a vector must change in a parity transformation? Isn't this just a special case of rotation, with the angle being 180? 2). Are Tom and Wrede actually defining two different types of vectors, or defining the same thing in two different ways?