Finding the Equation of a Plane from 3 Points

In summary, the conversation discusses finding the equation of a plane given three points. The suggested method is to take the cross product of two vectors from the points to find the normal vector, and then use the equation 0 = a(x-x1)+b(y-y1)+c(z-z1), with <a,b,c> being the normal vector. It is mentioned that it does not matter which point is used in the equation. Another approach is to write the equation as ax+by+cz=d, where <a,b,c> is the normal vector, and then solve for d by evaluating the equation at any point. Both methods will result in the same equation for the plane.
  • #1
Jeann25
30
0
I have a general question. If given 3 points, how would I find the equation of the plane containing all these points?
 
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  • #2
Do you have any ideas?

Do you know of any ways to write the equation of a plane given some other information? Can you gather that information if given 3 points?
 
  • #3
I think I found it. I would need to take the cross product of 2 vectors from those 3 points to find the normal vector. Then I would use the equation 0 = a(x-x1)+b(y-y1)+c(z-z1), <a,b,c> being the normal vector. For <x1,y1,z1> would I just pick one of the points? Does it matter which one?
 
  • #4
Your approach is fine and it doesn't matter which point you use in the end.
 
  • #5
That form of the plane equation works but I prefer to write the plane equation as ax+by+cz=d where <a,b,c> is a normal vector to the plane. You can then solve for d by evaluating the left-hand side at any point. In the end it doesn't matter because you'll end up with the same equation.
 
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FAQ: Finding the Equation of a Plane from 3 Points

What is the equation of a plane?

The equation of a plane is a mathematical representation of a flat, two-dimensional surface in a three-dimensional space. It is typically written as Ax + By + Cz + D = 0, where A, B, and C are constants and x, y, and z are variables.

How can I find the equation of a plane from three points?

To find the equation of a plane from three points, you can use the formula:
(x-x1)(y2-y1) - (x2-x1)(y-y1) = (x-x1)(z2-z1) - (x2-x1)(z-z1) = (y-y1)(z2-z1) - (y2-y1)(z-z1).
Plug in the coordinates of the three points into the formula and solve for A, B, C, and D.

Can I find the equation of a plane with only two points?

No, you need at least three non-collinear points to determine a unique plane in three-dimensional space. If you only have two points, there are infinitely many planes that can pass through them.

What if my points are not in a straight line?

If your points are not in a straight line, then they are non-collinear and you can use them to find the equation of a unique plane. However, if your points are collinear, then there are infinitely many planes that can pass through them and you will not be able to determine a unique equation.

Can the equation of a plane be written in different forms?

Yes, the equation of a plane can also be written in vector form
r = r0 + su + tv, where r0 is a point on the plane, u and v are two non-parallel vectors in the plane, and s and t are any real numbers. This form is useful for finding points on the plane or determining the intersection of multiple planes.

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