you're fine up to here.

suppose we pick x = 1. then 2z = 3x, means 2z = 3, and z = 3/2.So I think that should be

3

1

2

I'm sure my problem lies with how I arrange the numbers in the eigenvector answer, but I just don't know, it always seems to be the part I get wrong. Would someone mind explaining how to do it properly? Thanks in advance, I have been banging my head against this for days now and will be very thankful if someone can enlighten me!

then 2z = y means 2(3/2) = y, so that y = 3. this gives us the eigenvector (1,3,3/2). since any scalar multiple of this is also an eigenvector, we can multiply by 2 to clear out the fractions, giving us the eigenvector (for the eigenvalue λ = 14), 2(1,3,3/2) = (2,6,3), which is the same eigenvector your instructor got.