Convert from plane equation to scalar equation

[DespicableMe]

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.

 

[Mark44]

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 $\cdot$(x - P) = 0, where N is your normal, x = <x, y, z>, and P is the point you know.