Plane

       
           

We can define a plane as two dimensional surfaces which are flat in nature. When we work in two dimensional space we generally use the term plane instead of the two dimensional space. If a plane has more dimensions then we call it a hyper plane.
If there is an angle which intersects two planes then the angle is known as the dihedral angle.

In three dimensional spaces plane geometry will have following properties:

·         Two planes can be either parallel to each other, otherwise they will intersect each other.

·         In three dimensional plane, a line can be parallel to the plane or intersect the plane at a single point or can lie on the plane.

·         If there are two lines perpendicular to the same plane then lines will be parallel to each other.

·         If two planes are perpendicular to a line then plane must be parallel to each other.

We can define plane in geometry by three ways:

1:  If there is a three dimensional space and a normal vector then we can describe the plane in terms of position vector. Here r0 is the position vector and p0 is the plane, 'n' is vector normal to the plane then plane can be expressed as:
n. (r – r0) = 0.
Dot product of two perpendicular vectors is always zero.

2:  As we know that plane is a set of points then we can define the plane by a point and two vectors like:

r = r0 + sv + tw,

Here ‘s’ and ‘t’ is the range of real numbers, 'v' and 'w' are the vectors of plane and r0 is the position vector.

3:  Now in third method we will select three points and we will make the matrix of these points. We can find the plane by solving the matrix.

P1 = (x1, y1, z1), P2 = (x2, y2, z2) and P3 = (x3, y3, z3). 

Theorem (If Two Lines Intersect, Then exactly One Plane contains both Lines)

Theorems in math are generally proved using the already existing postulates. Therefore, we prove our theorem that if two Lines intersect, then exactly one plane contains both lines with help of some of the existing postulates as follows: The 1st postulate that we would use here is that a line can be considered only ...Read More

Math Topics