Vector

A vector is alternate notation for an n tuple, so that
\[
\begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix}
\]
is the n-tuple \( (a_1, a_2, \ldots, a_n ) \)

The length of a Vector

Given a vector in \( \mathbb{R}^n \), then we define the

dot product

Given two sequences of numbers of the same length: \( a = \left ( a_{1} , \ldots , a_{k} \right ) \) and \( b = \left ( b_{1} , \ldots , b_{k} \right ) \), their dot product denoted and defined by by \( a \cdot b = \sum_{i = 1}^{k} a_{i} \cdot b_{i} \). Note that \( \cdot : \mathbb{R}^{k} \times \mathbb{R}^{k} \to \mathbb{R} \)

vector form of a line

Suppose that \( d , p \) are vectors, then we say that the set \( l := \left \lbrace x : x = t d + p , \text{ for some } t \in \mathbb{R} \right \rbrace \) is a line and say that the vector equation for the line is

\( x = t d + p \)

We also say that \( d \) is the direction vector for \( l \)
An example of the vector form of a line in \( \mathbb{R}^{3} \) could be \( x = t \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] + \left [ \begin{matrix} 3 \\ 2 \\ 1 \end{matrix} \right ] \), where \( \left [ \begin{matrix} 3 \\ 2 \\ 1 \end{matrix} \right ] \) (the red dot) is a part of the line when \( t = 0 \), and the line's direction follows the direction of the vector \( \left [ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right ] \)