equation of planes

Implicit Form

A plane can be given by implicit equation

where and . This is called the scalar equation of planes.

In the above equation, the normal vectors are the multiple of

Alternatively, we can express the equation with scalar triple product

where are known points on the plane and is the parameter on plane) (this is called the vector equation of planes.

Derivation

To derive the above equation, we can use a normal vector and a point on the plane:

Alternatively, given three points , , and on the plane, we can construct the normal vector with the cross product: , so we can have an equation of plane as (or alternatively can be viewed as the determinant )

Importance in Multi-variable Calculus

Equation of planes is important in multi-variable calculus since when we go closer and closer to a surface, it can be approximated by a plane.

Parametric Form

A plane can also be expressed parametrically by a base point and two tangent vectors and :

See also

• equation of lines
• tangent space

Reference

• MIT OCW Multivariable Calculus > Session 8 Equations of Planes