# Definition:Vector Sum

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## Contents

## Definition

Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_n}$ and $\mathbf y = \tuple {y_1, y_2, \ldots, y_n}$ be vectors of an $n$-dimensional vector space.

Then the **(vector) sum** of $\mathbf x$ and $\mathbf y$ is given by:

- $\mathbf x + \mathbf y := \tuple {x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n}$

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.

If necessary, to distinguish the **vector sum** operation with other forms of addition, $+$ can be called **vector addition**.

## Also known as

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

## Also see

- If the vectors involved are written as column matrices, then
**vector addition**can be understood as matrix addition.

## Sources

- 1961: I.M. Gel'fand:
*Lectures on Linear Algebra*(2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Fundamental Definitions: $3.$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space - 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: $5 \ \text{(c)}$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.2$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**addition**(of vectors)