Search code examples
mathgraphics2dpointplane

finding implicit representation of plane

I have 2 questions:

  1. How can I find the implicit representation of a plane that passes through the point: (6,1,0), if it is given that the plane is parallel to a line that lies on the yz-plane
  2. Can we determine how many possible solutions exists?
Solution

  • The yz-plane has a simple basis, namely, {(0,1,0), (0,0,1)}. Take a point p in the plane you are looking for. Then p - (6,1,0) must lie on the yz-plane, i.e. it must be a linear combination of the basis. Therefore,

    p - (6,1,0) = λ(0,1,0) + μ(0,0,1)

    or

    p = λ(0,1,0) + μ(0,0,1) + (6,1,0)

    where λ and μ are the parameters.

    Note that the representation is not unique because it depends on the basis chosen for the yz-plane. A different basis would require different values for the parameters.