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Find the matrix for the transformation that projects each point in R3 (3-D) perpendicularly onto the plane 7x + y + 3z = 0 .

The attempt at a solution is attached for question 1 (actually instructor's solution)

I kind of understand it but ...

why is n <dot> v = equation of the plane?

Does v represent all of the possible points of R^3 (certainly does not seem so...) which is projected to the normal?

I understand the v-projection is there to get the projection of v onto the plane because we cannot directly project to the plane right? But why do we want v' what does v' represent and how is that the solution?

The attempt at a solution is attached for question 1 (actually instructor's solution)

I kind of understand it but ...

why is n <dot> v = equation of the plane?

Does v represent all of the possible points of R^3 (certainly does not seem so...) which is projected to the normal?

I understand the v-projection is there to get the projection of v onto the plane because we cannot directly project to the plane right? But why do we want v' what does v' represent and how is that the solution?