# 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 $\mathbb R^n.$

### Example

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

$$\begin{array}{rcl}Q&:=&(1,2)\\v&:=&\pmatrix{1\\2}\end{array}$$

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

$$v=\overrightarrow{OQ}:=\pmatrix{2-0\\1-0}.$$

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 $O.$

$$w=\overrightarrow{OP}:=\pmatrix{x_1-0\\x_2-0}.$$

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

$$v+w=\pmatrix{(2-0)+(x_1-0)\\(1-0)+(x_2-0)}=\pmatrix{2+x_1\\1+x_2}$$

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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