Equivalence of Definitions of Vector Sum

Proof
Let the Cartesian coordinate system be selected such that $\mathbf u$ and $\mathbf v$ are embedded in the $x$-$y$ plane.

Then:

and all other components, if there are any, are then zero.

Thus by the component definition:


 * $\mathbf u + \mathbf v = \tuple {u_1 + v_1, u_2 + v_2}$

Now we use the triangle law:


 * Vector-sum-equivalence.png

Let:

Thus we have:

It is seen that $\mathbf u + \mathbf v$ calculated using the triangle law is the same as $\mathbf u + \mathbf v$ calculated using the component definition.