What is the difference between a point and a vector? 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.

Since there is a trivial translation between vectors and the points they end at when based from the origin it`s a distinction without a difference in most contexts.
So if you want to be precise about it you can transform your points into the corresponding vectors, do your vector operations on those and transform them back to points again.
