Do the position and velocity vectors live in the same space? This is a question I've been thinking about for a while. Position vectors are supposed be represented by an arrow from the origin that traces out a path, while one does not need to think of the velocity vector as having any sort of starting point because as long as the length and direction and the same, they represent the same velocity. From a pure math perspective, a vector space is just an object with a dozen or so properties with closure being one of them. But it makes no sense to add a position and a velocity vector right? So that means they do not belong to the same vector space. But at the same time, it is possible to take the dot or cross products of a position and velocity vector. So how do you explain that?
Also when we learn multivariable calculus, why do we need the position vector to originate from the origin? $\Bbb R^3$ or $\Bbb R^2$ satisfies all the properties of a vector space without needing to conjure up any sort arrow.
 A: I think it's important to distinguish between the space of points, which is also known as affine space (let's denote this by $A$), and the space of vectors $V$. The space of points looks a lot like a vector space except there is no point called the origin, so an element in affine space is depicted as just a dot and not an arrow. A vector $v \in V$ is an arrow you can slide around in $A$. The assumption is that given any two points $p, q \in A$, there is a unique vector that starts at $p$ and ends at $q$.
You can indeed add a vector $v$ to a point $p$ by positioning the start of the vector at $p$ and defining $p+v$ to be the point at the end of the vector. You can also subtract two points $p$ and $q$ by defining $q-p$ to be the vector that goes from $p$ to $q$. But it makes no sense to add two points together.
However, if you choose a point $p_0$ in affine space, there is a unique vector from $p_0$ to each point $p$. This is the position vector of a point $p$ relative to $p_0$. So there is a natural isomorphism $I_{p_0}: A \rightarrow V$, where $I(p) = p-p_0$, identifying each point with a vector. The definition of the position vector depends on which point $p_0$ you choose to be effectively the origin of space.
So points and vectors live in different spaces, but the position vector does live in the same vector space as the vector itself.
