# Definition:Vector Quantity/Component

## Definition

Let $\mathbf a$ be a vector quantity embedded in an $n$-dimensional Cartesian coordinate system $C_n$.

Let $\mathbf a$ be represented with its initial point at the origin of $C_n$.

Let $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ be the unit vectors in the positive direction of the coordinate axes of $C_n$.

Then:

- $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_3 \mathbf e_n$

where:

- $a_1 \mathbf e_1, a_2 \mathbf e_2, \ldots, a_3 \mathbf e_n$ are the
**component vectors**of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ - $a_1, a_2, \ldots, a_3$ are the
**components**of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$.

The number of **components** in $\mathbf a$ is determined by the number of dimensions in the Cartesian coordinate system of its frame of reference.

A vector quantity with $n$ **components** can be referred to as an **$n$-vector**.

It is usually more convenient to write $\mathbf a$ as the ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ instead of $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_3 \mathbf e_n$.

There are two special cases:

### Cartesian Plane

Let $\mathbf a$ be a vector quantity embedded in a Cartesian plane $P$.

Let $\mathbf a$ be represented with its initial point at the origin of $P$.

Let $\mathbf i$ and $\mathbf j$ be the unit vectors in the positive direction of the $x$-axis and $y$-axis.

Then:

- $\mathbf a = x \mathbf i + y \mathbf j$

where:

- $x \mathbf i$ and $y \mathbf j$ are the
**component vectors**of $\mathbf a$ in the $\mathbf i$ and $\mathbf j$ directions - $x$ and $y$ are the
**components**of $\mathbf a$ in the $\mathbf i$ and $\mathbf j$ directions.

### Cartesian $3$-Space

Let $\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$.

Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.

Then:

- $\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$

where:

- $x \mathbf i$, $y \mathbf j$ and $z \mathbf k$ are the
**component vectors**of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions - $x$, $y$ and $z$ are the
**components**of $\mathbf a$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions.

It is usual to arrange that the coordinate axes form a right-handed Cartesian $3$-space.

## Einstein Summation Convention

Let $\mathbf a$ be a vector quantity.

$\mathbf a$ can be expressed in component form using the Einstein summation convention as:

- $\mathbf a = a_i \mathbf e_i$

## Also known as

The **components** of a vector quantity $\mathbf a$ as defined above can also be referred to as the **projections** of $\mathbf a$.

Some older sources refer to them as **resolutes** or **resolved parts**.

## Also see

## Historical Note

The idea of resolving a vector into $3$ **components** was originally due to RenĂ© Descartes.

## 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 - 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $4$. Components of a Vector - 1992: Frederick W. Byron, Jr. and Robert W. Fuller:
*Mathematics of Classical and Quantum Physics*... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.1$ Geometric and Algebraic Definitions of a Vector - 1992: Frederick W. Byron, Jr. and Robert W. Fuller:
*Mathematics of Classical and Quantum Physics*... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.2$ The Resolution of a Vector into Components