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 René 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