$l=\{\vec{x} : \vec{x} = t\vec{d}+\vec{p} \text{ for some } t∈\mathbb{R}\}$
Vector/Parametric form: $\vec{x} = t\vec{d}+\vec{p}$ $\text{ where } t∈\mathbb{R}$
($\vec{d}$ is a direction vector for $l$, $t$ is the parameter variable)
Coordinate: $\begin{bmatrix} x \\ y \end{bmatrix} = t \begin{bmatrix} d_1 \\ d_2 \end{bmatrix} + \begin{bmatrix} p_1 \\ p_2 \end{bmatrix}$
Parameter variables aren't quantities, they don't have to be equal to anything across
Slope-Intercept: $l=\{(x,y)∈\mathbb{R}^2: y=mx+b\}$
Ray: $R=\{\vec{x} : \vec{x} = t\vec{d}+\vec{p} \text{ for some } t\ge0\}$
Line Segment: $S=\{\vec{x} : \vec{x} = t\vec{d}+\vec{p} \text{ for some } t∈[0,2]\}$
Skew Lines: Non-parallel lines that don't intersect in 3D space.
Planes are defined by three points not on the same line $A,B,C∈\mathbb{R}^2$
A plane has infinitely many direction vectors, but only requires two to describe it.
Direction Vectors: $\vec{d_1}=\overrightarrow{AB}, \vec{d_2}=\overrightarrow{AC}$
Plane's Vector Form: $\vec{x}=\begin{bmatrix} x \\ y \\ z \end{bmatrix} = t\vec{d_1}+s\vec{d_2}+A$
$P=\{\vec{x}: \vec{x}=t\vec{d_1}+s\vec{d_2}+\vec{p} \text{ for some } t,s∈\mathbb{R}\}$
Note: Vector form must have direction vectors that are linearly independent!
Unit Square: $U=\{\vec{x}: \vec{x}=t\vec{e_1}+s\vec{e_2}+\vec{p} \text{ for some } t,s∈[0,1]\}$