# Is translation of position vectors allowed?

I have read that position vectors represent certain specific points in three dimensional space. So, therefore, translation of position vectors must not be allowed. Since, in any case it's done, it will represent a different position thereby violating its fundamental meaning. Furthermore, mathematically, this would also violate the representation of complex numbers using vector representation, since it would change the complex number.

I would like to have some insight into this matter. Thanks for an answer in advance.

• Read where? Also I'm not entirely clear what you mean by 'translating' a position vector. Commented Aug 27, 2020 at 14:20
• A position vector by definition denotes the displacement from an arbitrarily chosen origin to a point. The important point here is that the origin is arbitrarily chosen. Technically, to define the distance between two points we need a metric. $\Bbb R^n$ with just vector space structure does not have a notion of "the distance between the tips of two vectors" (it technically does not even have a notion of distance). Commented Aug 27, 2020 at 15:09
• Re, "representation of complex numbers using vector representation," Complex numbers are 2D vectors. That is to say, for every operation that is defined on 2D vectors, there is an isomorphic operation that is defined on complex numbers. The reverse is not true though. There are well defined operations on complex numbers (e.g., multiplication) for which there is no corresponding well-defined vector operation. nrich.maths.org/2432 Commented Aug 27, 2020 at 17:21

I have read that position vectors represent certain specific points in three dimensional space

They do not by themselves, as they are really displacement vectors from origin of coordinate system.

If you have point $$A$$ and "position" vector $$\vec{v}$$, then this defines new point $$B$$ by prescription $$B=A + \vec{v}$$, since classical space is really an affine space. You simply need an origin to get a point.

The "translation" of vectors is nothing but statement that in affine space the vector $$\vec{v}$$ can be added to whatever point and that all displacement vectors $$B-A$$ are from the same vector space. This is not so in curved space, where each vector lives in its own tangent space. I have recently written an answer to clarify how "translation" of vectors comes about from flat space special structure. Might be also of some relevance.

An absolute position vector represents the position of an object relative to a given origin. It is a bound vector because it depends on where we place the origin of our co-ordinate system - if we translate the origin but keep the object fixed then its position vector changes.

By contrast, a free vector is the same regardless of where the origin is. The position of one object relative to another object is a free vector. This is because it actually represents the difference of two bound vectors (the absolute position of each object is a bound vector). If we translate the origin then the change in one bound vector is offset by the change in the other, and the relative position vector does not change.

Another example of a free vector is the moment of a couple (a pair of equal and opposite forces acting at different points). This is the difference of two bound vectors (the moments of the two individual forces about the origin) so it is the same regardless of where we place the origin of our co-ordinate system.