# Definition:Vector Quantity/Component/Cartesian Plane

## Definition

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.

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

## 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.

## Historical Note

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