Use normals for parallel planes and lines inside a plane

(a) Plane Π passes through A=(2,0,0) parallel to $-2\mathbf i+\mathbf j+\mathbf k$ and $\mathbf i-3\mathbf j+2\mathbf k$.

(i) Find Π in scalar-product form and verify B=(6,−5,1) lies on it.3 marks

(ii) Find the position vector of the perpendicular foot from C=(−2,1,−3) to Π. 4 marks Hence find the equation of a parallel plane Π1 such that C is equidistant from Π and Π1.2 marks

(b) Line l and plane P are $l:\mathbf r=\lambda(\mathbf u+\mathbf v)$, λ real, and $P:\mathbf r\cdot(\mathbf u\times\mathbf v)=0$, with u,v nonzero and nonparallel.

(i) Show l is contained in P.2 marks

(ii) For u=(−1,2,1), v=(3,−3,2), find a line in P perpendicular to l through the origin.2 marks