Definition:Vector Addition/Vector Space

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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 $F$.


The group operation $+_G$ on $V$ is known as vector addition on $V$.


Vector Addition on Real Vector Space

The usual context for vector addition is on a real vector space or complex vector space:


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 $\mathbf u$.

Vector-sum-triangle-law.png

Then their (vector) sum $\mathbf u + \mathbf v$ is represented by the arrow $\vec {AC}$.


Sources

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