I am thinking specifically about $\mathbb{R}^2$ so I can visualize things.
By "origin" I mean that they start at the same point.
When we graphically represnt vectors we don't care where the starting point is (i.e. where the vector begins does not affect the vector; the vector $(1,2)^T$ is the same whether we draw it at the origin of $(10,10)$)
Since the point where we draw a vector starting from doesn't matter, Then I should be able to draw two different vectors that start at two different points.
Normally, if I want to graphically represent the angle between these two vectors, I would reposition them so that they start at the same point.Why is this the correct way to represent the angle between two vectors? (or maybe it isn't?)
That is, is there a part of the definition of the angle between two vectors that suggests that, graphically, we should represent them as originating from the same point?
Best Answer
Define vectors as directed line segments with equality between two vectors holding iff they point in the same direction and have the same length. You can now define the angle without insisting that the two vectors have the same origin.