What is the difference between a point and a vector?

Question

I understand that a vector has direction and magnitude whereas a point doesn't. However, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do cross product and dot product using two points instead of two vectors? I don't think so, but my roommate insists yes, and I'm kind of confused now.

Answer 1

There is a duality between points and vectors in the plane or in the space. A point is the mark of your pencil in the paper. A vector is an arrow from the origin of a coordinate system to the mark in the paper.

The duality works as follows. Given a point P there is one an only one vector v such that the tail is at the origin and the tip targets the point. Conversely, given a vector v place the origin of the vector system at the tail of the vector and define the point P as the tip of the vector.

From the dual point of view there is no difference between a point and a vector. They are the same object. Duality identifies marks in the paper with arrows in a given coordinate system. It is only two different ways to represent the same thing. For notation purposes, we represent points in parenthesis. For example P(1,2) and arrows in carets v=<1,2> in the first we mean the mark, in the second we mean the arrow. That's it!