# Straight Line in Space

## Point-Direction Form of the Equation of a Line

\[\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\]

where the point *P*_{1}(*x*_{1}, *y*_{1}, *z*_{1}) lies on the straight line and the vector **s** (*a*, *b*, *c*) is direction vector of the line.

## Two-Point Form of the Equation of a Line

\[\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}\]

## Equation of a Straight Line in Parametric Form

\[\left\{
\begin{aligned}
x &= {x_1} + t\cos\alpha \\
y &= {y_1} + t\cos\beta \\
z &= {z_1} + t\cos\gamma
\end{aligned}
\right.\]

where the point *P*_{1}(*x*_{1}, *y*_{1}, *z*_{1}) lies on the line and cos *α*, cos *β*, cos *γ* are the direction cosines of the direction vector of the line, the parameter *t* is any real number.