# Definition:Vector (Physics)/Component

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## Definition

A vector $\mathbf a$ can be represented with its initial point at the origin of a Cartesian coordinate system.

Let $\mathbf i, \mathbf j, \mathbf k$ be the unit vectors in the positive direction of the $x$-axis, $y$-axis and $z$-axis respectively.

Then:

- $\mathbf a = a_1 \mathbf i + a_2 \mathbf j + a_3 \mathbf k$

where:

- $a_1 \mathbf i, a_2 \mathbf j, a_3 \mathbf k$ are the
**component vectors**of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions - $a_1, a_2, a_3$ are the
**components**of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions.

The number of **components** in a vector is determined by the number of dimensions in the coordinate system of its frame of reference.

A vector with $n$ **components** is sometimes called an **$n$-vector**.

For a vector of more than three dimensions, the concepts of magnitude and direction are usually abandoned in favour of an ordered tuple of **components**.

## Also see

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Components of a Vector: $22.6$