【数学】 平面の方程式 基礎 Plane equation Basic

In 3D space of x, y, z,

A plane is defined by 3 points.

For example,

If you have a set of 3 points such as

A = (2, 4, 6)

B = (10, 2, 4)

C = (8, 8, 8)

then only one plane passing through the 3 points is identified.

But what is the equation of this plane?

You must find Normal vector (P, Q, R).

If you know the Normal vector,

the equation of the plane is

P (x - a) + Q (y - b) + R (z - c) = 0

, where (a, b, c) is a point on the plane.

For example, the equation of a plane with normal vector of (1, -7, 11) and passing a point (2, 4, 6) is,

(x - 2) - 7 (y - 4) + 11 (z - 6) = 0

But how do you find the normal vector, if all you are gven are 3 points?

Here is the answer.

The normal vector is a vector that is perpendicular to any 2 vectors formulated by the 3 points !!

For example,

The normal vector is not only perpendicular to

Vector AB =  (2, 4, 6) -->> (10, 2, 4)

but also perpendicular to

Vector AC = (2, 4, 6) -->> (8, 8, 8).

Vector (2, 4, 6) -->> (10, 2, 4)

is reduced to (8, -2, -2) === Vector AB

Vector (2, 4, 6) -->> (8, 8, 8).

is reduced to (6, 4, 2) === Vector AC

Then, how do you find the vector perpendicular to both of them?

This is the time to use cross product.

8   -2   -2     8   -2   -2
         X     X    X
6    4     2     6     4    2

-2x2 -  -2x4  = 4

-2x6 -   8x2  = -28

8x4   -  -2x6  = 44

Therefore, normal vector is (4, -28, 44)

This normal vector is perpendicular to the plane including points A, B, C.

Looking from the side...

Now you got the plane equation.

If this plane has normal vector of (4, -28, 44) and includes point A (2, 4, 6) as the following picture,

4(x - 2) - 28(y - 4) + 44 (z - 6) = 0.

Of course, this is the same plane with normal vector of (4, -28, 44) and includes point B (10, 2, 4).

4(x - 10) - 28(y - 2) + 44 (z - 4) = 0.

Of course, this is the same plane with normal vector of (4, -28, 44) and includes point C (8, 8, 8).

4(x - 8) - 28(y - 8) + 44 (z - 8) = 0.

Because the normal vector is perpendicular to

Normal vector (4, -28, 44) can be reduced to (1, -7, 11).

Therefore,

(x - 2) - 7(y - 4) + 11 (z - 6) = 0.

or

(x - 10) - 7(y - 2) + 11 (z - 4) = 0.

or

(x - 8) - 7(y - 8) + 11 (z - 8) = 0.

is also representing the same plane.