Adding Vectors
Two vectors add by doing one move after the other. Walk along
\vec{u}, and from wherever you end up, walk along
\vec{v}. The single arrow from your start to your finish is the
sum \vec{u} + \vec{v}. This is the
tip-to-tail rule: slide \vec{v} so its tail sits on
the tip of \vec{u}, and join up the ends.
In components it could not be simpler — just add the matching entries:
\begin{bmatrix} u_x \\ u_y \end{bmatrix} + \begin{bmatrix} v_x \\ v_y \end{bmatrix} = \begin{bmatrix} u_x + v_x \\ u_y + v_y \end{bmatrix}.
Tip to tail
Below, \vec{u} is drawn from the origin, then
\vec{v} is carried to its tip. The bold arrow back to the origin is
the sum. Drag the components and confirm the rule: the sum's entries are just the entries added.
Order doesn't matter
Walking \vec{u} then \vec{v} lands you in
exactly the same place as \vec{v} then
\vec{u} — the two routes are the two sides of a
parallelogram with the sum as its diagonal. So vector addition is
commutative, just like adding ordinary numbers:
\vec{u} + \vec{v} = \vec{v} + \vec{u}.
It is associative too, and adding the
\vec{0} vector changes nothing — the same friendly rules arithmetic
already taught you, now working on arrows.