stuckness...

All right. I hope the diagram is readable. I have several problems.
So, we have a vector, V, in the normal coordinate system (which has unit vectors i=(1,0,0) j=(0,1,0) and k-(0,0,1)). We want to bounce it off a wall and end up with vector V'. To make it easier, we define the coordinate system N', N, W with the origin at the point of intersection so that V is now in the plane M' which is shown above. The new coordinate system has unit vectors i'=(2/sqrt(5), 0, 1/sqrt(5)) j'=(0, 1, 0) and k'=(1/sqrt(5), 1, -2/sqrt(5)).

However, and here's the first question: doesn't the vector V have to be converted to the new system (making it V*) before any calculations can be done? I tried doing this two ways: ("*" is unfortunately also dot)

first, from the original packet:
V* = (V * i' , V * j', V * k')

next, as in the book:
V* = ( (i * i')Vx + (j * i')Vy + (k * i')Vz , (i * j')Vx + (j * j')Vy + (k * j')Vz , (i * k')Vx + (j * k')Vy + (k * k')Vz )

I tried the calculations for alpha and beta with both results. Only the first value for V* gave alpha and beta that passed the Pythagorean theorem test. So, second question: Why do I only use the way in the book for converting the V' back to the original coordinate system?

However, and this is kind of a side track, if alpha can be found by dotting -j' with V*, why can't beta be found by dotting -k' with V* ? I tried it, and it doesn't pass the Pythagorean test. Is it because V* is wrong, or is it just illegal somehow?

In any case, using the first way of finding V* and the ways of calculating alpha and beta that were in the last packet, I found V'=(0, 1, -sqrt(5)), and then used the way we worked out before based on the book to find V' in the original system and got V'=(1, 2, 4). The original vector in the original system was V=(1, -1, -2).
The length of V' isn't the same as V, the angle between them is not 2(theta) and V' isn't pointing the correct direction. So I know the answer isn't right, but I haven't been able to find what I'm doing wrong.