Separate a trigonometric parameter from a later special case

With respect to O, A,B,C have position vectors $\mathbf a=(1,\sin t,0)$, $\mathbf b=(\cos t,2,1)$, $\mathbf c=(2,-1,1)$, where $0\le t<\pi$.

(a) Find the exact t given a is perpendicular to b.3 marks

(b) X is on AB produced with AB:BX=1:4, and ACXY is a parallelogram. Take $t=\pi/2$ for the rest of the question.

(i) Find the position vectors of X,Y.4 marks

(ii) Find the area of ACXY. Hence find the shortest distance from X to line AC.3 marks

(iii) Line $l_1$ has equation $(x-2)/6=(3-y)/h$, z=k, where h,k are constants. The line through A,B intersects $l_1$ at a right angle. Find h,k.4 marks