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Equation of a plane

Definition. Plane is a surface containing completely each straight line, connecting its any points.

General form of the equation of a plane

Any equation of a plane can by written in the general form

A x + B y + C z + D = 0

where A, B and C are not simultaneously equal to zero.

Equation of the plane in segments

If the plane intersects the axis OX, OY and OZ in the points with the coordinates (a, 0, 0), (0, b, 0) and (0, 0, с), then it can be found using the formula of Equation of the plane in segments

x  +  y  +  z  = 1
a b c

Point-normal form of the equation of a plane

If you know the coordinates of the point on the plane M(x0, y0, z0) and the surface normal vector of plane n = {A; B; C}, then the equation of the plane can be obtained using the following formula.

A(x - x0) + B(y - y0) + C(z - z0) = 0

Describing a plane through three points

If given the coordinates of three points A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3), lying in a plane, the plane equation can be found by the following formula

x - x1y - y1z - z1  = 0
x2 - x1y2 - y1z2 - z1
x3 - x1y3 - y1z3 - z1

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