• Vector form of a plane equation
  • Coordinate form of plane equation
  • Intercept form of plane equation
  • Another vector form of plane equation
  • Parametric form of plane equation
  • Coordinate form of equation of plane in matrix form
• Example

Equation of plane

Let and be any two points in a plane and let be a vector normal to the plane. Since is perpendicular to , their dot product is equal to . The vector form of the plane equation through with normal is given by

Let

Substituting the above equations in (0.1),

where Equation (0.6) is the coordinate form, also called the rectangular Cartesian co-ordinate form of plane equation. The coefficients of in any equation of a plane are the components of vector normal to the plane.

Consider equation (0.6). It can be written as

Let

Then

Since in equation(0.9) are the intercepts of respectively, i.e. the plane intersects the coordinate axes at points , it is called the intercept form of the equation of plane.

Another vector form of equation of plane

Let be points in an affine space that are not collinear. Then the plane defined by is a set of points , where . Let be any general point in the plane. Then

is another vector form of equation of plane. Substituting the values in equation (0.10), we get the following set of equations

Parametric form of equation of plane

The parametric form of equation of plane through points and parameters is obtained by equating like-terms in equation (0.13).

Coordinate form of equation of plane in matrix form

In the figure above, is a vector perpendicular to the plane and lies on the plane.

Equation (0.17) is another vector form of equation of plane. Substituting the corresponding values in this equation, we get the coordinate form of equation of plane in the matrix form.

Example

1. Find the equation of plane that passes through (i) perpendicular to , (ii) parallel to the plane .
   
   1. Let be a point on the plane through and perpendicular to .

      

      are perpendicular to each other.

      

      is the equation of the plane through and perpendicular to ·

   2. If two planes are parallel, then both planes have the same normal vector. As we saw earlier, the co-efficient of i.e. is the component of the vector normal to the plane. Therefore, Equation of the plane is

      

      Since lies on the plane, the coordinates atisfy the equation of the plane. So solving for ,

      

      0.26 is the equation of the plane.

2. Given points find (i) the coordinate form of the equation of the plane through these points, (ii) parametric equation of the plane.
   1. Let be any point on the plane determined by .

      

      The coordinate form of equation of plane is obtzined by solving the equation below.

      

   2. Parametric form of equation of plane through is

      

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