I understand the process for creating a vector function r(t) that describes a line in 3D space. It is also my understanding that vectors can be picked up and moved around (provided I don't change their direction or length) and they are still the same vector. So my question is this:
If a vector function for a line generates an infinite number of vectors that "trace out" a 3D line, doesn't that same vector function also trace out every single line that is parallel to it in 3D space, since I can technically pick up all those vectors and originate them from any point in 3D space?
I know that when it comes down to the symmetric equations of a 3D line it wouldn't make sense, but in thinking about what a vector is and how if I say I have the vector <1, 2, 3>, it could be floating anywhere in space...is it just mathematically assumed that when we create vector functions for 3D lines/shapes/etc that they are all referring to vectors that originate at the origin point (0, 0, 0)?
There seems to be some (understandable) confusion about what a vector is.
This is because a "vector" has different meanings in different contexts.
In the basics, a "vector" is an element of a vector space. A prototypical example of a vector space is $\mathbb{R}^3$. Elements in $\mathbb{R}^3$ look like $(1,2,3)$ or $(-1,1,7)$, and you can draw them as arrows starting at $(0,0,0)$.
Now comes the confusing part: Vectors are often used to represent some quantities, such as force applied to an object, velocity of an object, flow of a fluid, etc. These vectors are actually tangent vectors, i.e. they belong to the tangent space of the coordinate system/manifold used to model your problem.
They still "start" at "$(0,0,0)$", but this is the $(0,0,0)$ of your tangent space, not the origin of your whole model. When people say that you can "move a vector around, and it is still the same vector", they mean it (I hope) in the same way as when I buy two loafs of bread, they are the same in the sense that they have the same size and nutritional value and taste etc., but they are not the same in the sense that if I eat one, the other not eaten by me.
Slightly more confusing still, is that if the coordinate system/manifold is just $\mathbb{R}^3$, people tend to identify the tangent space with the manifold itself.
To correctly describe a tangent vector, you should give both the base point and the direction. For example, you can consider the vector $(1,2,3)$ with base point $(3,4,5)$. The line described by this tangent vector is then given by $\mathbf{r}(t)=(3+1t,4+2t,5+3t)$. This is a different line than if the base point is different.