Mathguru – Help – Find the distance of a point from the given plane.

11.3 Planes and Angle Between Plane and Line > No. 14(b)

Find the distance of a point from the given plane.

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Explanation:

Plane (geometry)

Two intersecting planes in three-dimensional space

In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point (zero-dimensions), a line(one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

Planes embedded in ℝ³

This section is specifically concerned with planes embedded in three dimensions: specifically, in ℝ³.

Properties

In three-dimensional Euclidean space, we may exploit the following facts that do not hold in higher dimensions:

§ Two planes are either parallel or they intersect in a line.

§ A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.

§ Two lines perpendicular to the same plane must be parallel to each other.

§ Two planes perpendicular to the same line must be parallel to each other.

Distance from a point to a plane

For a plane $Description: \Pi : ax + by + cz + d = 0\,$ and a point $Description: \bold p_1 = (x_1,y_1,z_1)$ not necessarily lying on the plane, the shortest distance from $Description: \bold p_1$ to the plane is

$Description: D = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}}.$

It follows that $Description: \bold p_1$ lies in the plane if and only if D=0.

If $Description: \sqrt{a^2+b^2+c^2}=1$ meaning that a, b, and c are normalized then the equation becomes

$Description: D = \ | a x_1 + b y_1 + c z_1+d | .$

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Plane_(geometry)

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.

Planes embedded in ℝ3

Properties

Distance from a point to a plane

Planes embedded in ℝ³