# What are differences between affine space and vector space?

I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by Paul Bamberg and Shlomo Sternberg. In Chapter 1 authors define affine space and writes:

The space $\Bbb{R}^2$ is an example of a vector space. The distinction between vector space $\Bbb{R}^2$ and affine space $A\Bbb{R}^2$ lies in the fact that in $\Bbb{R}^2$ the point (0,0) has a special significance ( it is the additive identity) and the addition of two vectors in $\Bbb{R}^2$ makes sense. These do not hold for $A\Bbb{R}^2$.

Edit:

How come $A\Bbb{R}^2$ has point (0,0) without special significance? and why the addition of two vectors in $A\Bbb{R}^2$ does not make sense? Please give concrete examples instead of abstract answers . I am a physics major and have done courses in Calculus, Linear Algebra and Complex Analysis.

Consider the vector space $\mathbb{R}^3$. Inside $\mathbb{R}^3$ we can choose two planes, $P_1$ and $P_2$. The plane $P_1$ passes through the origin but the plane $P_2$ does not. It is a standard homework exercise in linear algebra to show that the $P_1$ is a sub-vector space of $\mathbb{R}^3$ but the plane $P_2$ is not. However, the plane $P_2$ resembles a $2$-dimensional vector space in many ways, primarily in that it exhibits a linear structure. In fact, $P_2$ is a classical example of an affine space.
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One defect of the plane $P_2$ is that it has no distinguished origin. One can artificially choose a point and redefine the algebraic operations in such a way to give it an origin, but that is not inherent to $P_2$. Another problem is that the sum of two vectors in $P_2$ is no longer in $P_2$. One can think of $AR^{2}$ as being modeled on this situation.