Line

We can express lines in ${\mathbb{R}}^3$ in vector form. Where ${\vec{r}}_0$ is an axis-intercept, $\vec{v}$ is a vector in the same direction (probably parallel to it) and $t\in \mathbb{R}$. In other words, a line is spanned by one vector.

$$L:\vec{r}={\vec{r}}_0+t\vec{v}=\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right]+t\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$$

We also have the corresponding parametric form of the line, where we basically just take the corresponding components:

$$x=x_0+at$$ $$y=y_0+bt$$ $$z=z_0+ct$$

The distance $d$ between some point $Q$ and the line is:

$$d=\frac{\Vert \vec{v}\times \overrightarrow{PQ}\Vert }{\Vert \vec{v}\Vert }$$

See also

• Plane
• Linear equation