Vector

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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 \( A = \begin{pmatrix}3 & 2 & -1\end{pmatrix}^T \), \( B = \begin{pmatrix}5 & 0 & -2\end{pmatrix}^T \), and define the direction vector from \( A \) to \( B \) as \( \vec{AB} = B - A = \begin{pmatrix}2 -2 -1\end{pmatrix}^T \).

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

Note: \( \vec{AB} \neq \vec{BA} \), in-fact \( \vec{BA} = - \vec{AB} \) in this case because they point in opposite directions.

We define the length of a vector \( \vec{A} \) as \( |\vec{A}| \), the true euclidean distance of the vector.