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【数学】 平面の方程式 Plane equation

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

https://remedics.air-nifty.com/photos/hawaii/graph10_2.gif

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


https://remedics.air-nifty.com/photos/hawaii/geogebra17.png

For example,

If you are given a formula of

https://remedics.air-nifty.com/photos/hawaii/graph11_2.gif,

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

https://remedics.air-nifty.com/photos/hawaii/geogebra18.png

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


https://remedics.air-nifty.com/photos/hawaii/geogebra19.png

It is (2, 1, 3).

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

https://remedics.air-nifty.com/photos/hawaii/graph11_2.gif,

it satisfies the equation.



This equation is sometimes expressed as a form of

https://remedics.air-nifty.com/photos/hawaii/graph15.gif,

or rarely in a form of

z = f (x, y)

as

https://remedics.air-nifty.com/photos/hawaii/graph14.gif

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.

https://remedics.air-nifty.com/photos/hawaii/geogebra20.png

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

https://remedics.air-nifty.com/photos/hawaii/geogebra21.png

https://remedics.air-nifty.com/photos/hawaii/geogebra22.png

In general, normal vector of a plane

https://remedics.air-nifty.com/photos/hawaii/graph10_2.gif

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

https://remedics.air-nifty.com/photos/hawaii/graph23.gif

It represents a line, and this line is perpendicular to

vector (A, B).



For example,

If you are given an eqation of

https://remedics.air-nifty.com/photos/hawaii/graph25_2.gif.

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

https://remedics.air-nifty.com/photos/hawaii/geogebra34.png

https://remedics.air-nifty.com/photos/hawaii/geogebra35.png

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

https://remedics.air-nifty.com/photos/hawaii/geogebra36.png.

https://remedics.air-nifty.com/photos/hawaii/geogebra37.png

The equation

https://remedics.air-nifty.com/photos/hawaii/graph25_2.gif

is the same as

https://remedics.air-nifty.com/photos/hawaii/graph3.gif.

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.

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