Definition:Vector Quantity/Component

Definition
A vector quantity $\mathbf a$ 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

 * Definition:Standard Ordered Basis