Straight line ($L$)

A straight line in Euclidean space is the set $L=\{x_0+tv:t\in \mathbb{R}\}$, for a fixed point $x_0$ and nonzero direction vector $v$. Two distinct points determine one such line.

A half-line beginning at $p$ with a nonzero direction vector $v$. Removing the endpoint gives an open ray; allowing every real parameter gives the whole straight line.

A ray with its endpoint removed. In the Argand plane, fixing the argument of $z-z_0$ specifies an open ray because the argument of zero is undefined.

A nonzero vector $d$ is a direction vector of a line if the line has a parametrization $r=p+td$, $t\in \mathbb{R}$, for some point $p$. Replacing $d$ by a nonzero scalar multiple describes the same line; dividing by its Euclidean norm gives a unit vector in one of the two orientations. A cross product of two nonparallel normal vectors supplies a direction vector for the intersection of their planes.

In Cartesian coordinates, the slope of a nonvertical straight line is its vertical change divided by its horizontal change. The graph $y=mx+c$ has slope $m$; a vertical straight line has no finite slope. For a tangent line to a differentiable function, the slope is the derivative.

A secant line is a straight line passing through two distinct points of a curve. For a graph $y=f(x)$ at $x=a$ and $x=a+h$, its slope is the difference quotient $(f(a+h)-f(a))/h$. As the second point approaches the first, a limiting secant line can give the tangent line.

A straight line through a point $a$ in nonzero direction $t$ is the set of vectors $r=a+st$ for real parameter $s$. If $|t|=1$, changes in the parameter equal signed distances along the line. Two such lines intersect exactly when their two parameter formulas give a common vector for some choice of their parameters.