A quantity that has both a magnitude (often representing distance or speed) and a direction. Graphically, a vector is represented in the Cartesian plane by an arrow with its root-end at the origin.

Figure 1: A vector in the Cartesian plane.

Notationally, a vector is usually represented by a variable symbol with an arrow drawn over it, and it is usually defined by an ordered tuple of numbers:

\[\vec{v}=\lt a_1,a_2\gt \]

The numbers in the tuple are called the components of the vector.

In Euclidean space the magnitude of the vector is found by using the general distance formula (taking the square root of the sum of the squares of the components). The magnitude of the example vector above would be given by:

\[\left| \vec{v}\right| =\sqrt{a_1^2+a_2^2} \]

Addition of vectors is componentwise, that is, the sum of two vectors \(\vec{v}=\lt a_1,a_2\gt\) and \(\vec{u}=\lt b_1,b_2\gt\) is given by:

\[\vec{v}+\vec{u}=\lt a_1+b_1,a_2+b_2\gt\]

Graphically, addition of vectors can be represented by completing a parallelogram whose sides are the vectors being added and whose diagonal is the sum.

Figure 2: The sum of two vectors.