Convert from plane equation to scalar equation

In summary, to convert an equation in the form r = r0 + sa + tb to scalar form Ax + By + Cz + D = 0, you first need to find the cross product of the vectors a and b, which will give you a normal to the plane. Then, use the formula N \cdot(x - P) = 0, where N is the normal, x is a point on the plane, and P is a known point, to express the equation in scalar form. This method works for both 2- and 3-dimensional planes.
  • #1
DespicableMe
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Homework Statement



When we're given an equation in form r = r0 + sa + tb, how do we convert it to scalar form Ax + By + Cz + D = 0?

I've been having trouble with plane form, but I can do it fine when its given in 2- or 3-D form.
 
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  • #2
DespicableMe said:

Homework Statement



When we're given an equation in form r = r0 + sa + tb, how do we convert it to scalar form Ax + By + Cz + D = 0?

I've been having trouble with plane form, but I can do it fine when its given in 2- or 3-D form.

In your vector equation, r = r0 + sa + tb, the vectors a and b are in the plane, so their cross product, a X b, would be perpendicular to the plane.

The vector r0 is a vector from the origin to a point on the plane.

Once you have a normal to the plane and a point on it, the equation of the plane is N [itex]\cdot[/itex](x - P) = 0, where N is your normal, x = <x, y, z>, and P is the point you know.
 

1. What is a plane equation?

A plane equation is a mathematical representation of a plane in three-dimensional space. It is typically written in the form of Ax + By + Cz + D = 0, where A, B, and C are the coefficients for the x, y, and z variables, and D is a constant value.

2. How do you convert from plane equation to scalar equation?

To convert from plane equation to scalar equation, you can simply isolate the z variable on one side of the equation. This will result in an equation in the form of z = mx + b, where m and b are coefficients determined by the values of A, B, C, and D in the original plane equation.

3. What is the purpose of converting from plane equation to scalar equation?

Converting from plane equation to scalar equation can be useful for graphing and visualizing a plane in two-dimensional space. It can also be helpful in solving problems that involve finding the intersection between a plane and another geometric object, such as a line or another plane.

4. Can any plane equation be converted to a scalar equation?

Yes, any plane equation in the form of Ax + By + Cz + D = 0 can be converted to a scalar equation. However, the resulting scalar equation may not always be in the form of z = mx + b, as it depends on the specific values of the coefficients A, B, C, and D.

5. Are there any limitations to converting from plane equation to scalar equation?

One limitation of converting from plane equation to scalar equation is that it only works for planes in three-dimensional space. Additionally, the resulting scalar equation will only represent one possible representation of the plane, as there are infinite ways to write a scalar equation for a given plane equation.

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