Can Cross Product Methods Yield Different Cartesian Equations for a Plane?

[choob]

this is just a simple question

For a plane in vector form, can the cartesian equation (Ax+By+Cz+D=0) be found by finding the cross product of the two vectors? My understanding is that A, B and C are the components of the normal of the plane, which can be found by doing the cross product. However, upon comparing my answers to those of a classmate, I discovered our answers were different, he used a method of substitution/elimination.

 

[Defennder]

Getting the normal to the plane isn't enough since any vector is normal to an infinite number of parallel planes. You must take the dot product of the plane normal vector with the position vector from a reference point on the plane to an arbitrary point on the plane (x,y,z) to get the equation. Your two answers should be the same, except that his may differ from yours by a multiple of the plane equation.

 

[Architeuthis Dux]

Yes, the cartesian equation of a plane can be found by finding the cross product of two vectors. The components of the normal vector, A, B, and C, represent the coefficients of x, y, and z in the cartesian equation. However, it is important to note that there are multiple ways to find the cartesian equation of a plane, and different methods may result in slightly different answers. It is important to double check your work and compare with others to ensure accuracy.