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Math Formulas: Planes in three dimensions
Plane forms
Point direction form:
where P1(x1,y1,z1) lies in the plane, and the direction (a,b,c) is normal to the plane.
General form:
where direction (A,B,C) is normal to the plane.
Intercept form:
this plane passes through the points (a,0,0), (0,b,0), and (0,0,c).
Three point form:
Normal form:
Parametric form:
where the directions (a1,b1,c1) and (a2,b2,c2) are parallel to the plane.
Angle between two planes:
is
The planes are parallel if and only if
Equation of a plane
The equation of a plane through P1(x1,y1,z1) and parallel to directions (a1,b1,c1) and (a2,b2,c2) has equation
The equation of a plane through P1(x1,y1,z1) and P2(x2,y2,z2), and parallel to direction (a,b,c), has equation
Distance
The distance of P1(x1,y1,z1) from the plane Ax + By + Cz + D = 0 is
Intersection
The intersection of two planes
is the line
where
If a = b = c = 0, then the planes are parallel.
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