# Definition:Vector Subtraction

## Definition

Let $\struct {F, +_F, \times_F}$ be a field.

Let $\struct {G, +_G}$ be an abelian group.

Let $V := \struct {G, +_G, \circ}_R$ be the corresponding **vector space over $F$**.

Let $\mathbf x$ and $\mathbf y$ be vectors of $V$.

Then the operation of **(vector) subtraction** on $\mathbf x$ and $\mathbf y$ is defined as:

- $\mathbf x - \mathbf y := \mathbf x + \paren {-\mathbf y}$

where $-\mathbf y$ is the negative of $\mathbf y$.

The $+$ on the right hand side is vector addition.

### Arrow Representation

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

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 $\vec {AC}$

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

Then their **(vector) difference** $\mathbf u - \mathbf v$ is represented by the arrow $\vec {CB}$.

## Also known as

The result $a - b$ of a **subtraction** operation is often called the **difference between $a$ and $b$**.

In this context, whether $a - b$ or $b - a$ is being referred to is often irrelevant, but it pays to be careful.

In some historical texts, the term **subduction** can sometimes be seen.

## Examples

### Example 1

Let:

\(\ds \mathbf a\) | \(=\) | \(\ds 6 \mathbf i + 4 \mathbf j + 3 \mathbf k\) | ||||||||||||

\(\ds \mathbf b\) | \(=\) | \(\ds 2 \mathbf i - 3 \mathbf j - 3 \mathbf k\) |

Then:

- $\mathbf a - \mathbf b = 4 \mathbf i + 7 \mathbf j + 6 \mathbf k$

## Also see

- Results about
**vector subtraction**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 - 1927: C.E. Weatherburn:
*Differential Geometry of Three Dimensions: Volume $\text { I }$*... (previous) ... (next): Introduction:Vector Notation and Formulae - 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.2$. Vector Spaces - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Fundamental Definitions: $3.$