The lines and 12 have equations r = 8i + 2j + 3k + 2(1 - 2j) and r = 5i + 3j - 14k + 12(2j - 3k) respectively. The point P on It and the point Q on I₂ are such that PQ is perpendicular to both I₁ and I₂. Find the position vector of the point P and the position vector of the point Q.

The points with position vectors 8i + 2j + 3k and 5i + 3j - 14k are denoted by A and B respectively.

Find

i. A̅P̅_(vector) × A̅Q̅_(vector) and hence the area of the triangle APQ,

ii. The volume of the tetrahedron APQB. (You are given that the volume of a tetrahedron is 1/3 x area of base x perpendicular height)