Definition:Vector Quantity/Component

Definition
Let $\mathbf a$ be a vector quantity embedded in an $n$-dimensional Cartesian coordinate system $C_n$.

Let $\mathbf a$ be represented with its initial point at the origin of $C_n$.

Let $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ be the unit vectors in the positive direction of the coordinate axes of $C_n$.

Then:
 * $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_3 \mathbf e_n$

where:
 * $a_1 \mathbf e_1, a_2 \mathbf e_2, \ldots, a_3 \mathbf e_n$ are the component vectors of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$
 * $a_1, a_2, \ldots, a_3$ are the components of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$.

The number of components in $\mathbf a$ is determined by the number of dimensions in the Cartesian coordinate system of its frame of reference.

A vector quantity with $n$ components can be referred to as an $n$-vector.

It is usually more convenient to write $\mathbf a$ as the ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ instead of $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_3 \mathbf e_n$.

There are two special cases:

Also see

 * Definition:Standard Ordered Basis