Equations of planes

Vector equation

The position vector \(\vec{r}=\langle x,y,z\rangle\) of a point on the plane is defined in terms of the position vector \(\vec{p}=\langle p_1,p_2,p_3\rangle\) of a known point on the plane, and two non-collinear direction vectors \(\vec{u}=\langle u_1,u_2,u_3\rangle\) and \(\vec{v}=\langle v_1,v_2,v_3\rangle\) in the plane. Two real parameters \(t\) and \(s\) are changed to get different points on the plane.

\[\vec{r}=\vec{p}+t\vec{u}+s\vec{v}\] \[\langle x,y,z\rangle=\langle p_1,p_2,p_3\rangle+t\langle u_1,u_2,u_3\rangle+s\langle v_1,v_2,v_3\rangle\]

Normal equation

Given the position vector \(\vec{p}\) of a known point \(P\) on the plane, and the normal vector \(\vec{n}\) to the plane, the position vector \(\vec{r}\) of any point \(P\) on the plane is related to them in the following way.

\[(\vec{r}-\vec{p})\cdot\vec{n}=0\]

Cartesian equation

The Cartesian equation of a plane has the form

\[Ax+By+Cz+D=0\]

such that \(\langle A,B,C\rangle\) is the normal vector to the plane and \((x,y,z)\) are the coordinates of any point on the plane.

Relation between two planes

  • If the normal vectors of the planes are parallel, then we have either one of the following cases.
    • The planes are co-incident (i.e. one and the same). It is sufficient to show that a point on one of them is also on the other.
    • The planes are parallel. They have no point in common. It is sufficient to show that a point on one of them is not on the other.
  • The planes are intersecting. Two planes intersect in a line. We can get the parametric equations of this line by solving the Cartesian equations of the planes.

Relation between three planes

  • If all normal vectors are parallel to one another, the planes may be either of the following.
    • Mutually co-incident (i.e. all one and the same).
    • Two co-incident planes and one parallel to them.
    • Three parallel planes.
  • If only two normal vectors are parallel, but not the third, the planes may be either of the following.
    • Two co-incident planes and one intersecting them.
    • Two parallel planes and one intersecting them.
  • If none of the normal vectors is parallel to another, the planes may be either of the following.
    • Pairwise intersecting; i.e. each pair of planes intersect at a line, but they have no common intersection. The three lines of intersection are parallel to one another.
    • Mutually intersecting at a line.
    • Mutually intersecting at a point.

Distance between a point and a plane

Given the Cartesian equation \(Ax+By+Cz+D=0\) of a plane and a point \(P(x_0,y_0,z_0)\), the distance \(D\) between them is calculated as

\[D=\dfrac{\lvert Ax_0+By_0+Cz_0+D\rvert}{\sqrt{A^2+B^2+C^2}}\]