Parallel Planes

A parallel planes are defined as a surface which is line joining of any two points on it lies completely on the surface. Parallel Planes are a horizontal and two-dimensional surface. Plane has two-dimensional analogue of a line (one-dimension), a dot (zero-dimensions), and a space (three-dimensions). Planes can occur as subspace of some upper dimensional space, as with a room wall, or they may like a separate own right in their continuation, as in the location of Euclidean geometry.

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Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or meet are called parallel lines.(Source: from Wikipedia)

Parallel planes and its various areas:

The parallel planes which do not intersect are known as parallel. One more planes especially in Hessian normal form is parallel if | n1. n2 | = 1 or n1 * n2 = 0.

Parallel  planes which are not parallel any time intersects in a line.

Parallel planes
Planes in various areas of mathematics:

Besides its familiar geometric structure, with isomorphism’s that are isometric with respect to the usual inner product, the plane may be viewed at various other levels of abstraction. Each intensity of abstraction corresponds with the specific categories.

At one extreme, all geometrical and metric concepts might be dropped to leave the topological plane, which may be thought of as an idealized phonologically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances.  The topological plane, or its equal of the open recording, is the basic topological neighborhood used to built surfaces (or 2-manifolds) classified in low-dimensional topology.

Example for parallel planes:

Examples:

Equation of the plane passing through a given point and parallel to two given vectors:

Let  `bara`   be the position vector of the given point A referred to the origin O.

Let `u` and `v` be the given vectors, which are parallel to the plane.

Let `p` be any point on the plane and let its position vector be `r`  (i.e.,) = `op` = `r` .

Through A, draw a lines AB and AC parallel to `u` and  `v`  lying in the planes such that `AB` =   `u`   and  `AC` = `v`

Now `AP` is coplanar with `AB` and `AC` .

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Therefore  `AP` = s `AB` +t `AC` Where s and t are scalars

=  `su` + `tv`

= `OP` + `OA` + `AP`

`=>`   `r` = `a` + `s`  `u` + `tv`

This is the vector equation of the plane (in parametric form).