【数学】 平面の方程式 Plane equation

A plane that goes through a point (a, b, c) is expressed as,

where (A, B, C) is a normal vector of this plane.

For example,

If you are given a formula of

,

this describes a plane floating in the xyz 3D world.

Can you tell which point does this plane go through by looking at the equation?

3 (x - 2) + 4 (y - 1) + 6 (z - 3) = 0

It is (2, 1, 3).

Because if you put (x, y, z) = (2, 1, 3) into the formula of

,

it satisfies the equation.

This equation is sometimes expressed as a form of

,

or rarely in a form of

z = f (x, y)

as

But if you successfully manipulate the equation and found the Coefficient (A, B, C) of x, y, z, those are the component of Normal Vector of the plane.

Normal vector (A, B, C) represents a vector perpendicular to the plane's surface.

For example,

If you are given a formula of

3 (x - 2) + 4 (y - 1) + 6 (z - 3) = 0

A vector (3, 4, 6) is "perpendicular" to the plane of

In general, normal vector of a plane

is described as

n = (A, B, C).

Rule:

Any plane can be defined, if you have a point (a, b, c) going through, and a normal vector (A, B, C)

This principle can be applied to line.

If you saw the equation like

It represents a line, and this line is perpendicular to

vector (A, B).

For example,

If you are given an eqation of

.

The line is passing through a point (1, 3).

And perpendicular to vector (4, -3). .

.

The equation

is the same as

.

This form of line can tell that the line also goes through a point (-2, -1),

and parallel to

vector (3, 4).

Why? ---> Line formula, advanced.