Definition: Points vs. Vectors in a Number Space

Let $P=(x_1,\ldots,x_n)$ and $Q=(y_1,\ldots,x_n)$ be two points of an $n$-dimensional number space $\mathbb R^n.$ Then the point $V=(v_1,\ldots,v_n)\in\mathbb R^n$ defined by the componentwise difference $v_1=x_1-y_1,\ldots,v_n=x_n-y_n$ is identified with a vector $$v:=\overrightarrow{PQ}=\pmatrix{v_1\\\vdots\\v_n}=\pmatrix{x_1-y_1\\\vdots\\x_n-y_n}.$$ In particular, any point $P=(x_1,\ldots,x_n)$ can be identified with the vector $$\overrightarrow{OP}:=\pmatrix{x_1-0\\\vdots\\x_n-0},$$ where $O$ is the origin point of $\mathbb R^n.$


This definition gives us the possibility to calculate with points as if they were vectors, i.e. to interpret the number space $\mathbb R^n$ with the vector space $\mathbb R.$ Consider the following example for $n=2:$ A point $Q$ and a vector $v$ in a $2$-dimensional number space $\mathbb R^2$ could be denoted by


Although both objects, the point $Q$ and the vector $v,$ have the same coordinates, they denote different things in mathematics and in physics.

The vector $v$ can be interpreted as the difference between the endpoints $Q$ and the origin $O.$ We can write this relationship as


This difference preserves the length and the direction of vectors identified with points, regardless of the position of these points. In order to see it, consider another vector $w$ defined as the difference of the endpoint $P$ and the origin $O.$


If we add this vector $w$ to the vector $v$ using the rules of calculation in the vector space $\mathbb R^2,$ then we get the vector


While the coordinates of the point $(2+x_1,1+x_2)$ denote another position, this does not change the length and direction of the original vector $v$. You can try it by moving the point $P$ from the origin to another position in the above interactive figure.

Thus, while some points and vectors have the same coordinates in a number space, they are different - vectors are invariant with respect to translations, points are not. This can be written as $$\overrightarrow{OP}+\overrightarrow{PQ}=\overrightarrow{OQ}.$$

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