How to find condition of three planes intersecting at a point (according to vector approach)?

Question

For example, if

Plane 1: $2x+y+z=1$

Plane 2: $x-y+z=2$

Plane 3 : $4x-y+3z =5$

How to check if these planes intersecting at a single point? I want to check the coordinates of intersection , from scalar triple product or solving from normals of the planes .

Answer 1

You just have to solve the linear systems formed by the equations of your planes, and to see whether it has solutions or not.