How do I determine the point of intersection of three planes?

Answer 1

A plane in three - space has the equation ax + by +cz = d

Where at least one of the numbers a,b,c must be non-zero.

There are three ways that three planes can intersect.

1)The planes intersect a point .There is exactly one solution.

(The normals are not parallel and not coplaner)

2)The planes intersect in a line.There are an infinite number of solutions.

(The normals are coplaner but not parallel)

3) The planes are coincident .There are infinite number of solutions.

(The normals are parallel)

There are three posibilities for a system of three planes with no solution.

i)The planes are parallel and atleast two are distinct.

ii)Two planes are parallel and third intersect both of the parallel planes.

iii)The planes intersect inpairs.

Normal can be analysed to determine the type and number of solutions to a system of three planes.

The three planes are

There are three equations and three unknowns.you may use substitution or elimination to solve the system.

We use also matrix to determine the point of intersection of three planes .

Solving matrix equations of the form AX = B

A is coefficient matrix, X is variable matrix and B is constant matrix.

Camers rule, Gaussian elimination method  and inverse matrix method.