Vector

Tags

math

Vectors are collections of values organised into an array. They can be used to represent a coordinate-position (position vector) or a direction that something is pointing in (direction vector).

For example you can have 2 position vectors $A={\left(\begin{array}{ccc}3& 2& -1\end{array}\right)}^T$, $B={\left(\begin{array}{ccc}5& 0& -2\end{array}\right)}^T$, and define the direction vector from $A$ to $B$ as $\overrightarrow{AB}=B-A={\left(\begin{array}{c}2-2-1\end{array}\right)}^T$.

Note: The ${}^T$ superscript above means transposed. Position and direction vectors are commonly expressed in column notation (such as $\left(\begin{array}{c}5\\ 0\\ -2\end{array}\right)$) however for brevity and readability their transposed and written as row vectors. The distinction will become more relevant as we explore matrices.

Note: $\overrightarrow{AB}\ne \overrightarrow{BA}$, in-fact $\overrightarrow{BA}=-\overrightarrow{AB}$ in this case because they point in opposite directions.

We define the length of a vector $\overrightarrow{A}$ as $|\overrightarrow{A}|$, the true euclidean distance of the vector.