# Origin of vectors with physical meaning

I was reviewing notes of physics, and i realized that something about the mathematics of vectors was wrong in my head.

Suppose a vector is $\vec{A}=5\textbf{i} + 3\textbf{j}$, and other $\vec{B}=7\textbf{i}+3\textbf{j}$. Then $\vec{A}-\vec{B}=\vec{C}=-2\textbf{i}$.

Now, take for example a point charge vector field:

$${\mathbf {E}}({\mathbf {r}})= \ q_{i}{\frac {{\mathbf {r}}-{\mathbf {r}}_{i}}{|{\mathbf {r}}-{\mathbf {r}}_{i}|^{3}}}$$ (Dropped the constants)

This kind of vectors have a new implementation, they are attached to a point in space.

I do not know if it is the same with velocity vector.

I just want to know if the idea is wrong and what should I review.

• The $\vec{r}$ is position vector of a point in space. Although vectors themselves are not affected by translation, but fixed vectors like the position vector have various applications like specifying the position of a point wrt a fixed point (origin). Consider the case of force being applied on a rigid body, in that case the point of application of force does matter, as it may produce some torque. So when it comes to application of vectors, we may fix their tails in some situations. Same is the case for velocity of a rolling rigid body. May 19, 2017 at 14:02
• If i put r (position vector) or i put x,y,z (if i am dealing with cartesian coordinates) is the same? I mean, it is the point what is important.. And something else.. Vectors are not attached to a point in space, but physicists do that for practical purposes?
– user153036
May 19, 2017 at 14:31
• It seems like your question depends on coordinate systems - vectors are independent of your choice of coordinate system, you can think of them as representing a change from a reference point. So if your vector adds 2 to the x position, in one system you can go from x to x+2 and (x-1) to (x+1) and the vector is the same because the displacement is too May 19, 2017 at 14:35
• Actually $P(x,y,z)$ is the coordinate, but $\vec{r}$ is position vector of that coordinate. Thus $\vec{r} = x\hat{i}+y\hat{j}+z\hat{k}$. Vectors are sometimes treated as being fixed, in physics as I earlier said, and in mathematics too, for example consider the equation of line in 3 dimensions ($\vec{r} = \vec{r_o} + \lambda \vec{b}$). Here, $\vec{r}$ and $\vec{r_o}$ are taken to be fixed as they tell about location of a point in a coordinate system. $\vec{b}$ is not fixed, as it is used to scale along the line. May 19, 2017 at 14:40
• Mathematically, you're looking for the concept of affine space. This may help you imagine things better. May 19, 2017 at 16:06

In purely mathematical sense, you can put the origin of the electric field vector $\textbf{E}$ everywhere in the space. This is ok, actually.