Plane/Vector space/Intuitive/Example

Let ${}E$ be a "plane“ with a fixed "origin“ ${}Q\in E$. We identify a point ${}P\in E$ with the connecting vector ${}{\overrightarrow {QP}}$ (the arrow from ${}Q$ to ${}P$) In this situation, we can introduce an intuitive coordinate-free vector addition and a coordinate-free scalar multiplication. Two vectors ${}{\overrightarrow {QP}}$ and ${}{\overrightarrow {QR}}$ are added together by constructing the parallelogram of these vectors. The result of the addition is the corner of the parallelogram which lies in opposition to ${}Q$. In this construction, we have to draw a line parallel to ${}{\overrightarrow {QP}}$ through ${}R$ and a line parallel to ${}{\overrightarrow {QR}}$ through ${}P$. The intersection point is the point sought after. An accompanying idea is that we move the vector ${}{\overrightarrow {QP}}$ in a parallel way so that the starting point of ${}{\overrightarrow {QP}}$ becomes the ending point of ${}{\overrightarrow {QR}}$.

In order to describe the multiplication of a vector ${}{\overrightarrow {QP}}$ with a scalar ${}s$, this number has to be given on a line ${}G$, which is also marked with a zero point ${}0\in G$ and a one point ${}1\in G$. This line lies somehow in the plane. We move this line (by translating and rotating) in such a way that ${}0$ becomes ${}Q$, and we avoid that the line is identical with the line given by ${}{\overrightarrow {QP}}$ (which we call ${}H$).

Now we connect ${}1$ and ${}P$ with a line ${}L$, and we draw the line ${}L'$ parallel to ${}L$ through ${}s$. The intersecting point of ${}L'$ and ${}H$ is ${}s{\overrightarrow {QP}}$.

These considerations can also be done in higher dimensions, but everything takes place already in the plane spanned by these vectors.