Definition:Vector Quantity/Component

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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 René Descartes.


Sources

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