vector

in mathematics, a vector is a quantity that cannot be expressed by a single number (called a scalar)

Euclidean vector

usually, a vector refers to a Euclidean vector (sometimes called a geometric vector or spatial vector)
we just call Euclidean vectors vectors

a vector is a geometric object that has magnitude and direction, represented as a directed line segment, or graphically as an arrow connecting initial point $A$ and terminal point $B$

a vector can be shown as a lowercase letter with an arrow on top

for example, let vector $\vec{a}$ have initial point $A$ and terminal point $B$
this can be represented with $\overrightarrow{AB}$, and its magnitude is denoted by $|\overrightarrow{AB}|$

Euclidean vectors can be added and scaled to form a vector space

row and column vectors

matrices with a single row are called row vectors, and those with a single column are called column vectors

for example, the column vector $\left[\begin{array}{c}3\\ 2\end{array}\right]$ represents a vector $\overrightarrow{AB}$ where $A$ is at $(0,0)$ and $B$ is at $(3,2)$. the magnitude of this vector can be calculated using trig: $|\overrightarrow{AB}|=\sqrt{3^2+2^2}$

position vector

a point in n-dimensional space, $A$, can be represented relative to an origin point $O$ using a position vector - a column vector with $n$ rows