Planes

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 Sub Topics Planes along with points and lines form the building blocks of Geometry. These are named as undefined terms as they are explained only using examples and descriptions and not with definitions. In algebra you are familiar with the coordinate plane on which points are plotted using ordered pairs and lines are represented as equations. Thus, a plane can be described as a flat surface made up of infinitely many points extending in all directions without thickness..A plane can be represented by a capital letter like Plane X. It can also be represented using at least three non collinear points that lie on the plane, like plane ABC.

Plane Equation

Planes are represented by algebraic equations in 3D-Geometry. The 3D-space is illustrated using 3 coordinate axes, x, y and z. The coordinate planes are names as xy, yz and xz planes named after the axes contained by the planes as shown in the diagram.

The 3-D equation to XY - plane is z = 0.
Similarly the respective equations to YZ and XZ planes are x = 0 and y = 0 .
The equations of planes parallel to coordinate planes are as follows:
z = k is the equation of a plane parallel to xy - plane at a signed distance k from it.
x = k is the equation of a plane parallel to yz - plane at a signed distance k from it.
y = k is the equation of a plane parallel to xz - plane at a signed distance k from it.
A first degree equation in variables x, y and z represent a general plane in 3-D space.
ax + by + cz + d = 0.

Different formulas both vector and algebraic are available to write the equation of a plane based on given conditions.

1. Equation to a plane containing the point P (x0, y0, z0) and perpendicular to the direction $\overrightarrow{n}$ =
The vector form of the equation is n . r = n . r0.  where r0 is the position vector 0, y0, z0>.
The 3-D form of the equation is a(x - x0) + b(y - y0) + c(z - z0) = 0.
Combining the constant terms, the equation to the plane can be written as ax + by + cz + d = 0.
2. Equation to a plane passing through a given point P(x0, y0, z0) and parallel to two vectors
$\overrightarrow{u}$ = 1, m1, n1> and  $\overrightarrow{v}$ = < l2, m2, n2>
The vector form of the equation is
$\overrightarrow{r}$ =  $\overrightarrow{a}$ + s $\overrightarrow{u}$ + t $\overrightarrow{v}$
where $\overrightarrow{a}$ is the position vector of P.
The algebraic form of the equation is $\begin{vmatrix} x-x_{0}&y-y_{0} & z-z_{0}\\ l_{1}& m_{1} & n_{1}\\ l_{2}& m_{2} & n_{2} \end{vmatrix}$ = 0
3. Equation of a plane containing two given points P(x1, y1, z1) and Q(x2, y2, z2) and parallel to a given vector         $\overrightarrow{v}$ = .
The vector form of the equation is $\overrightarrow{r}=(1-s)\overrightarrow{a}+s\overrightarrow{b}+t\overrightarrow{v}$
where $\overrightarrow{a}$ and $\overrightarrow{b}$ are position vectors of the points P and Q and t and s are
parameters.
The algebraic form of the equation is $\begin{vmatrix} x-x_{1} & y-y_{1} &z-z_{1} \\ x-x_{2} &y-y_{2} &z-z_{2} \\ l & m & n \end{vmatrix}$ = 0
4. The equation of the plane passing through three given points P(x1, y1, z1), Q(x2, y2, z2) and R(x3, y3, z3).
The vector form of the equation is
$\overrightarrow{r}$ = $\overrightarrow{a}$ + s($\overrightarrow{b}$ - $\overrightarrow{a}$) + t($\overrightarrow{c}$ - $\overrightarrow{a}$)  where $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ are the position vectors
of the points P, Q and R.
The algebraic form of the equation is $\begin{vmatrix} x-x_{1} & y-y_{1} &z-z_{1} \\ x-x_{2}& y-y_{2} &z-z_{2} \\ x-x_{3}&y-y_{3} & z-z_{3} \end{vmatrix}$ = 0

Coplanar

Three points define a plane and hence they are coplanar. If more than three points lie on the same plane, they are called coplanar points. To check whether a given point is coplanar with other points, the coordinates of the point should satisfy the equation obtained for the other points.
Two lines define a plane and hence they are coplanar. If a plane can be found containing more than two lines, then they are called coplanar lines. For a line to lie on a given plane, its direction should be perpendicular to the normal vector of the plane.

Distance From a Point to a Plane

The distance D from the point P(x1, y1, z1) to the plane ax + by + cz + d = 0 is given by the formula,
D = $\frac{|ax_{1}+by_{1}+cz_{1}+d|}{\sqrt{a^{2}+b^{2}+c^{2}}}$You may note that this formula is similar to the formula used for finding the distance from a point to a straight line in 2-D Geometry.

Intersection of Planes

The intersection of two or three planes gives the following possibilities:
1. All the planes are parallel. They do not intersect. (For both the case of two or three planes)
2. All the planes may be coincident. (For both the case of two or three planes)
3. The planes may intersect in a line. (For both the case of two or three planes)
4. Each of the planes intersect the other planes in separate lines. (In the case of three planes)
5. The planes intersect in a point ( In the case of three planes)

Let us look at the cases of intersection two and three planes separately.

Intersection of Two planes

Two planes will intersect, if their normal directions are not equal.
When two planes have the same normal vector \$\overrightarrow{n},then the two planes are parallel and hence do not intersect.
If two planes intersect, they intersect on a straight line. Algebraically this is equivalent to saying a system of two equations in three variables having infinite solutions. (All the points on the intersecting line).

To find the line of intersection of two intersecting planes, the equations can be solved algebraically to get the parametric equation of the line of intersection.
The other situation of intersecting planes is that the two planes are coincident. This is the case when the equation of one line is a constant multiple of the equation of the other line.

Intersection of Three Planes

In the case three intersecting planes, we have four possibilities.
1. The three planes intersect at a straight line.  From algebraic perspective, this is the case a system of three equations
in three variables having infinite solutions (All the points on the line of intersection).

2. The three planes intersect in a point. This is equivalent to a system of three equations in three variables having a
unique solution.
3.  Each of the plane intersect the other two planes separately in a straight line.

4. Two planes are coincident and the third plane intersecting them in a straight line.
5.  All the three planes are coincident.

Line Plane Intersection

When we consider a line and a plane, there are three possibilities:
1. The line may lie entirely on the plane. In this case, both the line and the plane will have the same normal directions.

2. The line may cut the plane in a point.

3. The line never intersects the plane. In this case as well, both the line and plane will have the same normal directions.