# Definition:Vector Sum

## Definition

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

### Component Definition

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

 $\ds \mathbf u$ $=$ $\ds \tuple {u_1, u_2, \ldots, u_n}$ $\ds \mathbf v$ $=$ $\ds \tuple {v_1, v_2, \ldots, v_n}$

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 right hand side is conventional addition of numbers, while the $+$ on the left hand side takes on a different meaning.

The distinction is implied by which operands are involved.

### Triangle Law

Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that:

$\mathbf u$ is represented by $\vec {AB}$
$\mathbf v$ is represented by $\vec {BC}$

that is, so that the initial point of $\mathbf v$ is identified with the terminal point of $\mathbf u$.

Then their (vector) sum $\mathbf u + \mathbf v$ is represented by the arrow $\vec {AC}$.

## Also known as

A vector sum is also frequently seen referred to as a resultant.