The three vectors are: $$a_1=i+j+k$$ $$a_2=i-j$$ $$a_3=i+j-2k$$
I have been asked to express the vector $b=2i-3j+5j$ in terms of the three vectors above like: $b= \alpha a_1+\beta a_2+\gamma a_3$. However, I cannot seem to be able to do it. If it helps, the set of vectors form an orthogonal set. I suspect that determinants may be involved because that's what the lectures were covering (which is what this problem was set for as part of a problem sheet). I just have no idea how to do it. Any help would be appreciated.
$$b=xa_1+ya_2+za_3=(x+y+z)i+(x-y+z)j+(x-2z)k=2i-3j+5k.$$ Then you have 3 equations with $x,y,z$ from here you should do it. $$x+y+z=2$$ $$x-y+z=-3$$ $$x-2z=5$$