Definition:Vector Quantity/Component/Cartesian 3-Space

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Definition

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.


It is usually more convenient to write $\mathbf a$ as the ordered tuple $\tuple {x, y, z}$ instead of $\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$.


$x$ Component

The value $x$ is known as the $x$ component of $\mathbf a$.


$y$ Component

The value $y$ is known as the $y$ component of $\mathbf a$.


$z$ Component

The value $z$ is known as the $z$ component of $\mathbf a$.


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.


Examples

Velocity

Let $\mathbf v$ be a velocity of a particle $P$ in space.

Then we have:

$\mathbf v = \dfrac {\d \mathbf r} {\d t}$

where $\mathbf r$ is the displacement of $P$ at time $t$.

Thus:

$\mathbf v = \dfrac {\d x} {\d t} \mathbf i + \dfrac {\d y} {\d t} \mathbf j + \dfrac {\d z} {\d t} \mathbf k$

where:

$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$


Acceleration

Let $\mathbf a$ be an acceleration of a particle $P$ in space.

Then we have:

$\mathbf a = \dfrac {\d \mathbf v} {\d t} = \dfrac {\d^2 \mathbf r} {\d t^2}$

where:

$\mathbf v$ is the velocity of $P$ at time $t$
$\mathbf r$ is the displacement of $P$ at time $t$.

Thus:

$\mathbf a = \dfrac {\d^2 x} {\d t^2} \mathbf i + \dfrac {\d^2 y} {\d t^2} \mathbf j + \dfrac {\d^2 z} {\d t^2} \mathbf k$

where:

$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$


Also see


Historical Note

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


Sources

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