A line can be defined as a Vector equation given by a starting vector and a direction vector. Note that the vector equation applies to any dimension of .
Definition
Vector Line ^formula
\vec{r} = \vec{r}_{0} + t\times \vec{d}
^terms
* $\vec{r}_{0}$ = The initial vector that the line passes through
* $\vec{d}$ = The direction vector which determines the slope of the line, and which the line is parallel to
* $t$ = A scalar used to parameterise the line, $t \in \mathbb{R}$
##### Parametric Form
We can also represent the vectors $\vec{r}, \vec{r}_{0}$ and $\vec{d}$ as components. Assuming the vector line is in [3D Real Space](3D%20Real%20Space.md), we can use $x,y,z$:
(x,y,z) = (x_{0},y_{0},z_{0}) + t(a,b,c)
\begin{cases} x = x_{0} +ta, & (1)\ y = y_{0} + tb, & (2)\ z = z_{0} + tc, & (3) \end{cases}
##### Cartesian Form
If $\vec{d}$ has non-zero components in all it's dimensions, i.e. $a, b,c \neq 0$, then we can solve the parametric system for $t$.
\begin{cases} t = \dfrac{x-x_{0}}{a}, & (1)\ t = \dfrac{y-y_{0}}{b}, & (2)\ t = \dfrac{z-z_{0}}{c}, & (3) \end{cases}
\dfrac{x-x_{0}}{a} = \dfrac{y-y_{0}}{b} = \dfrac{z-z_{0}}{c}
>[!info] In the cases where $a,b$ or $c$ equal to zero
>
>We can still get the parametric form, but simply need to not equate all the coefficients. For example, if $c = 0$
>$$
>\dfrac{x-x_{0}}{a} = \dfrac{y-y_{0}}{b}, z= z_{0}
>$$
### Geometry
Obviously, the vector equation returns a line. But $\vec{r}_{0}$
#TODO
### Pair of Lines
In a space containing two vector lines, $\vec{r}_{1}$ and $\vec{r}_{2}$ in 2D Space, $\mathbb{R}^2$ there are only two possibilities:
* Both lines are parallel: They have the same direction vectors, but their initial vectors are different
* Both lines intersect at some point: In this scenario, the direction vectors must be different, and they have different time values for when they intersect.
However, if the lines are in [3D Real Space](3D%20Real%20Space.md), $\mathbb{R}^3$, a third possibility exist:
* Both lines are skewed: They do not intersect nor are they parallel.