Hi attached question and mark scheme answer . I understand everything up to part (d)

I don't follow how using [ λ ,1/2-1/2 λ , 3-λ] dot product [2,-1,-2]=0 is used . Are these two vectors perpendicular .

I understand it evaluates to λ=13/9 and gives the coordinates .

I am also struggling with the alternative answer of differentiation for OP^2.

Thank you