VECTORS
A vector is a quantification of magnitudes in two or more dimensions. Each dimension refers to a specific attribute being considered, while a magnitude for a given dimension refers to the measurement of the specific attribute the dimension represents. Note that the units of measurement may be the same or may vary with dimensions. When notating a vector, the mapping between the magnitudes and the dimensions must be clear, which is why vectors are notated as ordered lists of values with index referring to dimension.
To help illustrate this concept, consider the attributes as dimensions of space. For now, consider two-dimensional (2D) space. To be consistent, the horizontal axis shall be the first element and the vertical axis shall be the second element. Then, the vector $\left[1 \atop 2 \right]$ represents the movement of $1$ along the horizontal axis, then $2$ along the vertical axis. It can be visualised as an arrow in a space with defined axes. In general, any $n$-dimensional vector can be visualised - or at least conceptualised - as a motion or an end-point with respect to an origin in an $n$-dimensional space. For convenience, when conceptualising vectors in this way, the default origin of the vector shall be the origin of an $n$-dimensional cooridnate space.