Projections onto a plane and parallel to a vector

[Blacky&Imy]

Hi. I'm preparing the Linear algebra 1 Exam for the first year of
Physics University. I find very difficult to understand projections.
Here's an example:
Consider the projection P: R^3 --> R^3 onto the plane U of equation
2x1 - 3x2 + x3 = 0 and parallel (the projection) to the vector v=(2,1,0).

This is to say, P is the linear application defined as follows:

P(x1,x2,x3) = ( -3x1 + 6x2 - 2x3 , -2x1 + 4x2 - x3 , x3 )

I can't see why P is defined this way, how they got it, so I would
appreciate if you could show me the passages step-by-step. In general, I can't understand how to define a projection, given the vector and the plane. Also, I'd
like to understand what does "parallel to a vector" means in this case.

Thanks a lot.

 

[Bacle2]

There are many sources out there; you start by projecting into the normal vector to the plane:

http://metamerist.blogspot.com/2007/03/vector-projection-on-plane.html