Equations of Lines and Planes

Lines and Planes

A line is described by a point on the line and its angle of inclination, or slope. Every line lies in a plane which is determined by both the direction and slope of the line. A line is essentially a representation of a cross section of a plane, or a two dimensional version of a plane which is a three dimensional object.

Equations of Lines and Planes

The components of equations of lines and planes are as follows:

A line in three dimensional space is given by a point, $P_0 (x_o,y_o,z_o)$, or a plane, $M$, and its direction. This direction is described by a vector, $\mathbf{v}$, which is parallel to plane and $P$ is the arbitrary point on plane $M$.

The position vector of point $P_0$ is called $\mathbf{r}_0$ and the position vector of point $P$ is called $\mathbf{r}$. The vector from $P$ to $P_0$ is called vector $\mathbf{a}$. Vectors $\mathbf{a}$ and $\mathbf{v}$ are parallel to each other.

Now, we can use all this information to form the equation of a line on plane $M$. 

The vector equation of a line is:

$\mathbf{r} = \mathbf{r}_0 + t \mathbf{v}$

where $t$ represents the location of vector $\mathbf{r}$ on plane $M$.

The parametric equation of a line can be written as: 

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

Or the more compact form:

 $<x, y, z> \ = \ <x_0 + at, y_0+bt, z_0 + ct>$

Vertical Line, Graphed

Vertical line $x = a$, lying on the $xy$-plane ($z=0$).