Addition of Vectors?

Question

How do you explain in general the addition of any two vectors geometrically without reference to any coordinate system?

Answer 1

Another way is to remember that vectors describe translations, that is moving without rotating. The arrow just shows for a single point where that point ends up after the movement. The addition of the two vectors is just the result of doing both translations one after another; the resulting translation can then again be described by an arrow that starts at an arbitrary point, and ends where that point ends up after doing that combined movement.

This method has the advantage that it doesn't matter whether the arrows start at the same point, and that it is obvious that it doesn't matter where the arrow starts; all arrows with the same length and direction describe the same vector, because they describe the same translation.

Also from this, one naturally gets the common description of vector addition by starting the second arrow at the end of the first one; the natural explanation being "this point is moved to there by the first translation, and then further to over there by the second translation; the sum arrow goes from the starting point directly to the final end point".

Answer 2

If the starting point of $\vec{A}$ and $\vec{B}$ are coincident, $\vec{A} + \vec{B}$ is the vector from that common starting point to the far corner of the parallellogram which has as two of its sides $\vec{A}$ and $\vec{B}$.

There is no need for a coordinate system in this definition. The coordinate system comes about when you start saying things like $\vec{A} = <2,3>$. ($2$ what? $3$ what?)

Answer 3

It is sufficient to check wikipedia (http://en.wikipedia.org/wiki/Parallelogram_law):