Definition:Vector Sum/Component Definition

Definition
Let $\mathbf u$ and $\mathbf v$ be vector quantities of the same physical property.

Let $\mathbf u$ and $\mathbf v$ be represented by their components considered to be embedded in a real $n$-space:

Then the (vector) sum of $\mathbf u$ and $\mathbf v$ is defined as:
 * $\mathbf u + \mathbf v := \tuple {u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n}$

Note that the $+$ on the is conventional addition of numbers, while the $+$ on the  takes on a different meaning.

The distinction is implied by which operands are involved.

Also see

 * Equivalence of Definitions of Vector Sum