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Equation of a Plane

Date: 05/09/97 at 02:01:54
From: Aparna Yelamarti
Subject: Formulas Of Planes and Spheres

Can you please give me the formula of a plane when:

   1) Any three points on it are given.

   2) A point is given and the equation of the normal is known.


Date: 05/09/97 at 09:27:35
From: Doctor Mitteldorf
Subject: Re: Formulas Of Planes and Spheres

Dear Aparna,
Say you know the direction of a normal to a plane is a vector (a,b,c).  
This means that traveling in the direction (a,b,c) gets you away from 
the plane as fast as possible. Conversely, the expression 
ax + by + cz is constant everywhere on the plane. You can evaluate 
the constant by substituting the one point you know on the plane, and 
then you have an equation for the plane.
Sometimes you're not given the direction vector (a,b,c) for the line, 
and instead you know TWO equations for the line: ax + by + cz = r and
dx + ey + fz = s. There are two equations for a line in 3-D because 
one equation specifies a plane, and two equations specify the line 
where they intersect. If you take the vector cross product of the two 
vectors (a,b,c) and (d,e,f), that should give you the vector that 
points in the direction of the line.
If you know three points on a plane and you want an equation for the
plane, you can take the difference between point 1 and point 2 and 
call that the first vector. Then take the difference between point 1 
and point 3 and call that the second vector. Now if you take the 
cross product of the first vector and the second vector, you will have 
a vector that points normal to the plane. (This is because it is a 
fundamental property of the vector cross product that the result is 
always perpendicular to the two vectors you are multiplying.)  Once 
you have the normal and any one point, you can proceed as above.    

-Doctor Mitteldorf,  The Math Forum
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Associated Topics:
College Higher-Dimensional Geometry

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