In linear algebra, a Euclidean vector, or simply a vector, is a 1-dimensional matrix. For example, a column vector with $m$ elements is an $m \space \text{x} \space 1$ matrix, consisting of a single column. For example,
$$ \vec{x} = \left[ \begin{matrix} x_1 \\ x_2 \\ \vdots \\ x_m \\ \end{matrix} \right] $$
Similarly, row vector is a $1 \space \text{x} \space n$ matrix for some $n$ , consisting of a single row of $n$ entries,
$$ \vec{a} = [a_1 \space a_2 \space \dots \space a_n ] $$
Vectors are directed line segments that have both a magnitude and direction. The direction of the vector is represented by the start and end point of the line and the magnitude of the vector is represented by the length of the line.
The norm of a vector refers to the length or the magnitude of a vector. For a vector $\vec{x}$, its norm is denoted as $\|\vec{x}\|$
There are different measures for the norm. The norm of a vector is a non-negative value. A norm must satisfy the following properties:
Unit Vector
The unit vector has a magnitude of 1. Given a non-zero vector $\vec{x}$, its normalised vector is the unit vector in the same direction, i.e.
$$ \hat{x} = \frac{\vec{x}}{\|\vec{x}\|} $$
Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the standard unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are
$$ \hat{x} = \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right], \hat{y} = \left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right], \hat{z} = \left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right] $$
They form the standard basis of the three-dimensional space which is a set of mutually orthogonal (mutually perpendicular) unit vectors.