Find a perpendicular foot on a parameterised line

With respect to O, A,B,C have position vectors $\mathbf a=5\mathbf i+4\mathbf j+10\mathbf k$, $\mathbf b=-4\mathbf i+4\mathbf j-2\mathbf k$, $\mathbf c=-5\mathbf i+9\mathbf j+5\mathbf k$.

(i) Find the position vector of D such that ABCD is a parallelogram.2 marks

(ii) N divides AB in the ratio $\lambda:(1-\lambda)$. Show that $\overrightarrow{CN}=(10-9\lambda)\mathbf i-5\mathbf j+(5-12\lambda)\mathbf k$. Hence find the position vector of N given CN is perpendicular to AB.4 marks

(iii) P has position vector $(-1+3t)\mathbf i+4\mathbf j+(2+4t)\mathbf k$, where $t\in\mathbb R$. Show that P is collinear with A and B for all real t.3 marks