Definition:Vector Quantity

Definition
A vector quantity is a is a real-world concept that needs for its model a mathematical object with more than one component to specify it.

Formally, a vector quantity is an element of a vector space, often the real vector space $\R^n$.

The usual intellectual frame of reference is to interpret a vector quantity as having:
 * A magnitude
 * A direction.

This can be informally interpreted as "something that points in some direction".

This can be rendered on the page like so:


 * Vector.png

In a Euclidean $n$-space $\R^n$, it is implied that the arrow issues from the origin:
 * $O = \underbrace {\tuple {0, 0, \ldots, 0} }_n$

Alternatively, and frequently more usefully, a 'vector quantity can also expressed in terms of coordinates.

In the above diagram, this would be the "head" of the 'vector quantity.

It is important to note that there is no mathematical difference between interpreting a 'vector quantity in $n$-space as "just the tip of the arrow" or "an arrow issuing from $O$ ending at the tip of the arrow". It is only a manner of connotation: both an arrow and a point have the same defining property of an ordered tuple.

In the contexts of physics and applied mathematics, it is a real-world physical quantity that needs for its model a mathematical object which contains more than one (usually numeric) component.

In this context it is frequently referred to just as a vector.

Also see

 * Definition:Scalar Quantity
 * Definition:Scalar (Vector Space)
 * Definition:Scalar Field