# 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.