# 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**.

## Also see

- Definition:Matrix Entrywise Addition: for when the vectors involved are written as column matrices

- Results about
**vector addition**can be found**here**.

## Sources

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Addition and subtraction of vectors: $4$. Component and Resultant - 1966: Isaac Asimov:
*Understanding Physics*... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Forces and Vectors - 1972: M.A. Akivis and V.V. Goldberg:
*An Introduction to Linear Algebra & Tensors*(translated by Richard A. Silverman) ... (previous) ... (next): Chapter $1$: Linear Spaces: $1$. Basic Concepts - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $5 \ \text{(c)}$ - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.2$