Vector ( i-j = k) ?

That's wrong!
[itex]
(i-j)\times (i+j)=\underbrace{i\times i}_0+i \times j- j\times i-\underbrace{ j \times j}_0=i\times j+i\times j=2 i \times j=2 k
[/itex]

 

It's wrong before that, even.
##\vec P = P(\vec j -\vec k)/\sqrt 2##
Alternatively, by calculating the distance from A to the line of P the moment is seen to have magnitude ##aP\sqrt {\frac 32}##. I believe the correct answer is exactly half that shown.

 

-j X i=k
That k and the other normal one from i x j=k, add up to 2k.