Find a projection foot using two methods

Relative to O, A,B,C have position vectors $-3\mathbf i+8\mathbf j+\mathbf k$, $7\mathbf i+3\mathbf j+6\mathbf k$, and $\alpha\mathbf i+\beta\mathbf j-6\mathbf k$ respectively, where α,β are real.

(i) Given $BC:\mathbf r=(7+3\lambda)\mathbf i+3\mathbf j+(6+4\lambda)\mathbf k$, find α,β.2 marks

(ii) Find the length of projection of AB onto BC.2 marks

(iii) Hence or otherwise find the position vector of the perpendicular foot from A to BC.3 marks

(iv) Find the exact area of triangle ABC.3 marks