Cartesian equation of a plane

In summary, the question is whether the cartesian equation of a plane can be found by taking the cross product of two vectors. The answer is no, as simply finding the normal vector of the plane is not sufficient. The equation can be obtained by taking the dot product of the normal vector with the position vector from a reference point to an arbitrary point on the plane. Two answers may differ by a multiple of the plane equation.
  • #1
choob
32
0
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.
 
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  • #2
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.
 
  • #3


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.
 

What is the Cartesian equation of a plane?

The Cartesian equation of a plane is a mathematical representation of a plane in three-dimensional space using the coordinates x, y, and z.

How is the Cartesian equation of a plane written?

The Cartesian equation of a plane is typically written in the form ax + by + cz + d = 0, where a, b, and c are the coefficients of x, y, and z, respectively, and d is a constant.

How do you find the Cartesian equation of a plane?

To find the Cartesian equation of a plane, you need to know three points on the plane or the normal vector of the plane. Using these points or vector, you can determine the values of a, b, c, and d in the equation ax + by + cz + d = 0.

What does the Cartesian equation of a plane represent?

The Cartesian equation of a plane represents all the points in three-dimensional space that lie on that plane. It can also be used to determine the distance between a point and the plane, and to find the intersection of two planes.

What is the significance of the coefficients in the Cartesian equation of a plane?

The coefficients a, b, and c in the Cartesian equation of a plane represent the direction cosines of the normal vector of the plane. This means that they determine the orientation and slope of the plane in relation to the x, y, and z axes.

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