Concept of Plane in 3 Dimensional Geometry

Typically, a plane is a two-dimensional flat surface with a capacity to be extended infinitely. It is a surface on which two points are joined to make a line segment, then the line segment completely lies on the surface itself. In the space that is 3-dimensional, a plane forms an equation- ax+by+cz+d=0 

• One must be non-zero from a, b, and c for the equation to exist. 
• Any straight line that seems perpendicular to every line lying on the plane, is commonly known as the normal to that plane. 
• Every normal to a plane is basically parallel to each other.
• There are some conditions for the plane to be uniquely determined. If any among the given is certainly known, the plane’s equation can be determined:

1. The plane’s normal with its distance from the origin is given i.e., the plane’s equation in normal form.
2.  The given plane is progressing through three non-collinear points. 
3. The given plane seems perpendicular to a particular direction and progresses through a point.

Equation of a plane when parallel to the coordinate planes

The plane’s equation will be z=c when the xy-plane is parallel to the given plane.

The plane’s equation will be  x=a when the yz-plane is parallel to the given plane.

The plane’s equation will be y=b when the zx-plane is parallel to the given plane.

Plane’s equation in normal form

Let’s consider a plane whose perpendicular distance from its origin is d, and the unit vector to the plane is n. Such plane’s equation in vector form will be

        r.n=d                                – (1)

Here the position vector is represented by r on the plane and n denotes the normal (unit) vector. For calculating the unit normal vector, we can use n=nn .

Now, let’s find out the plane’s equation in Cartesian form.

Let’s consider a point Q (x, y, z) on the plane. Then, this point will be r=xi+yj+zk  distant from the origin. 

Now, let’s the direction cosines of normal unit vectors be l, m, and n. So, n=li+mj+nk.

On putting the respective values in equation 1, we get

xi+yj+zk.li+mj+nk=d

                             lx+my+nz=d                        – (2)

The above equation is the plane’s equation in the normal form (in Cartesian form).

Note: Considering the vector form of the plane’s equation in the normal form i.e., r.n=d, if normal is not in the form of a unit vector, then to convert it into unit vector, we shall divide both sides of the equation by n.

Equation of a Plane when it is perpendicular to a given vector and progressing through a given point

Usually, many planes are there that seems perpendicular to any given vector, but if there is any given point  A(x1,y1,z1), only one such plane exists.

Consider a plane progressing through a given point A with a as its position vector and n being the given vector to which it is perpendicular. The plane’s equation in vector form is

r–a.n=0

Here, the position vector of any point (x, y, z) on the plane is r.

Now, consider a plane that is progressing through a given point A(x1,y1,z1) and (a, b, c) be the direction ratios of the line to which it is perpendicular. The Cartesian form of such plane’s equation is 

ax-x1+by-y1+cz-z1=0

This is also known as one point form for the equation of the plane.

Equation of Plane passing through three given points that are non-collinear

Consider a plane coursing through three given points that seem to be non-collinear namely R, S, and T having position vectors a, b, and c, respectively. The equation of such plane is vector form is

r–a.b–ac–a=0

Here, the position vector of any point P on the plane is r.

Now, consider a plane progressing through given three non=collinear points that are R, S, and T with coordinates (x1,y1,z1) , x2,y2,z2, and (x3,y3,z3) respectively. The equation of such plane in Cartesian form is 

=0

Here P is any given point on the plane.

Conclusion

The concept of a plane in 3-D geometry is finding the equation of such a plane in different contexts. An equation of a plane in space can be determined if the plane passes through three non-collinear points, if the plane progresses through a point and it is perpendicular to a particular known direction, or if its distance from the origin and the normal to the plane is given. There are many other concepts of the equation of a plane in 3-dimensional geometry, such as the plane’s equation in its intercept form, a plane’s equation that is parallel to a given plane etc.