Distance of a Point from a Plane

Let us see about distance of a point from a plane,
In mathematics, a plane is placed at distance from a point of any horizontal, two-dimensional smooth surface. A plane is the two dimensional have a distance of a point that means zero-dimensions, a line that is known as one-dimension and a space  is known as three-dimensions. A lot of mathematics may be performed in the surface of plane, particularly in the areas of geometry, trigonometry, graph theory and graphing.

Properties of plane:

• Two planes may be parallel or both intersect in a line.
• A line may be parallel to a plane, intersects in point.
• Two lines at a 90 degree angle to the same plane should be parallel to each other.

• Two planes at a 90 degree angle to the similar line should be parallel to each other.

Point - Plane distance:

Let us see about distance of point from plane:
Given a plane,


and a point X0 = (xo,y0,z0),the normal to the plane is given by,

and a vector from the plane to the point is given by,

Projecting W onto V gives the distance D from the point to the plane as,

This equation is known as the distance of a point from a plane.

Examples:

1)      Find the distance from the point P = (2, 2, 4) for the plane 2x +2y + 3z + 4 = 0.
Solution:
We use formula from the distance of a point from a plane.
From the above equation we substitute for the plane A = 2, B = 2, C = 3, D = 4. From the point P, we substitute x₁ = 2, y₁ = 2, and z₁ = 4.
Plane's distance of a point is,


2) Find the distance from the point P = (2, 3, 5) for the plane x - y + z + 5 = 0.
Solution:
We use formula from the distance of a point from a plane.
From the above equation we substitute for the plane A = 1, B =- 1, C = 1, D = 5. From the point P, we substitute x₁ = 2, y₁ = 3, and z₁ = 5.
Plane's distance of a point is,