Finding the Equation of a Plane from 3 Coplanar Points

In summary, the method for determining the equation of a plane involves finding the cross product of two vectors on the plane to get the normal vector, and then substituting a point into the Cartesian equation to solve for d. The coefficients of the normal vector represent the x, y, and z components that make up the equation of the plane. This is because the dot product of any vector in the plane with the normal vector must be equal to 0, resulting in the equation ax+by+cz=(ad+be+cf).
  • #1
DiamondV
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Homework Statement


The method that we are taught on how to determine the equation of a plane is as follows when given 3 coplanar points:
1.
Determine the vectors
2.
Find the cross product of the two vectors.
3.
Substitute one point into the Cartesian equation to solve for d.

Homework Equations

The Attempt at a Solution



I know how to do this but my issue is with the intuition behind it, by getting the cross product of two vectors on the plane we are essentially getting the normal vector of the entire plane. we then take the coefficents of this vector and put it into sort of an equation like this x+y+z=d, then sub a point into this to find d and that's how you get the equation of the plane. I mean what exactly is happening here? How does the coefficents of the x, y and z components(or magnitude of x,y,z, basically whatever is front of the x, y,z) give us the equation of the plane?
 
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  • #2
The cross product of two vectors in the plane will be a vector ##\vec v=(a,b,c)## that is normal to the plane. Let ##D=(d,e,f)## be any point in the plane. Then for any other point ##P=(x,y,z)## in the plane, the vector ##\vec u## from D to P lies in the plane and hence must be perpendicular to ##\vec v##.

So we have

$$0=\vec u\cdot \vec v =(x-d,y-e,z-f)\cdot(a,b,c)=a(x-d)+b(y-e)+c(z-f)=ax+by+cz-(ad+be+cf)$$

So the equation of the plane is

$$ax+by+cz=(ad+be+cf)$$
 

1. How do you find the equation of a plane from 3 coplanar points?

To find the equation of a plane from 3 coplanar points, you can use the method of cross products. First, determine the vectors between each pair of points and take the cross product of two of these vectors. This will give you the normal vector of the plane. Then, use one of the given points and the normal vector to form the equation of the plane in the form of ax + by + cz = d.

2. What are coplanar points?

Coplanar points are points that lie on the same plane. This means that they can be connected by a straight line and do not require any bending or stretching to be placed on the same plane. In other words, the points are in the same two-dimensional space.

3. Can you find the equation of a plane with only 2 coplanar points?

No, to find the equation of a plane, you need at least 3 coplanar points. This is because 2 points can only form a straight line, which does not uniquely define a plane.

4. What is the significance of the normal vector in finding the equation of a plane?

The normal vector is perpendicular to the plane and is used to define the orientation of the plane. It is also used in the equation of the plane to determine the coefficients of x, y, and z.

5. Are there other methods to find the equation of a plane from 3 coplanar points?

Yes, there are other methods such as using the formula for the distance between a point and a plane or using the components of the normal vector. However, the method of cross products is the most commonly used and straightforward method.

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