Equations of lines

Vector equation

The position vector \(\vec{r}=\langle x,y,z\rangle\) of any point on the line is defined in terms of the position vector \(\vec{p}=\langle p_1,p_2,p_3\rangle\) of a known point on the line and a direction vector \(\vec{v}=\langle v_1,v_2,v_3\rangle\) that is parallel to the line. A parameter \(t\) is a real number that can be changed to obtain a different point on the line.

\[\vec{r}=\vec{p}+t\vec{v}\] \[\langle x,y,z\rangle=\langle p_1,p_2,p_3\rangle+t\langle v_1,v_2,v_3\rangle\]

Parametric equations

If we equate each component on the left side of the vector equation with its counterpart on the right side, we get the parametric equations, which separate the components \(x\), \(y\) and \(z\).

\begin{align} x=p_1+tv_1;\qquad y=p_2+tv_2;\qquad z=p_3+tv_3 \end{align}

The parameter \(t\) is the same for the \(x\), \(y\) and \(z\) for any given point on the line.

Symmetric equations

The symmetric equations are obtained by re-arranging the parametric equations for \(t\).

\[\dfrac{x-p_1}{v_1}=\dfrac{y-p_2}{v_2}=\dfrac{y-p_3}{v_3}\]

If a component of \(\vec{v}\) is zero, the corresponding part of the symmetric equations does not exist, because it would result in division by zero. The symmetric equations of a line in 2D are easily re-arranged into either the standard form \(Ax+By+C=0\) or the slope-intercept form \(y=mx+b\).

Relations of lines in 3D

  • If the direction vectors are parallel (scalar multiples of one another), the lines are either one of the following.
    • Co-incident (i.e. one and the same). We only need to show that a point on one of them is also on the other. The lines have all their points in common.
    • Parallel (no points in common). We only need to show that a point on one of them is not on the other.
  • If the direction vectors are not parallel, the lines are either one of the following.
    • Intersecting. They have only one point in common. The system of parametric equations is consistent; i.e. has a solution.
    • Skew. They have no points in common. The system of parametric equations is inconsistent; i.e. does not have a solution.

Relation of a line to a plane

  • The direction vector of the line is perpendicular to the normal vector of the plane. This means either of the following cases.
    • The line is parallel to the plane; i.e. they have no point in common. It is enough to show that a point on the line is not on the plane.
    • The line is contained in the plane; i.e. all points of the line are also on the plane. It is enough to show that a point on the line is also on the plane.
  • The line intersects the plane at one and only one point.

Distance between a point and a line

In 2D

Given the standard form \(Ax+By+C=0\) of the equation of the line, and the coordinates \((x_0,y_0,z_0)\) of the point, the distance \(D\) is calculated as

\[D=\dfrac{\lvert Ax_0+By_0+C\rvert}{\sqrt{A^2+B^2}}\]

In 3D

Given the direction vector \(\vec{v}\) of the line and a known point \(P_0\) on the line, the distance \(D\) between point \(P\) and the line is calculated as

\[D=\dfrac{\lVert \vec{PP_0}\times \vec{v}\rVert}{\lVert \vec{v}\rVert}\]