Category:Vector Subtraction

This category contains results about Vector Subtraction.
Definitions specific to this category can be found in Definitions/Vector Subtraction.

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 $\vec {CB}$.