# Definition:Vector Quantity/Component/Cartesian Plane

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## 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**.

## Examples

### Example

Component of Vector in Plane/Examples/Example 1

## Also see

## Historical Note

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

## Sources

- 1957: D.E. Rutherford:
*Vector Methods*(9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 1$. - 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