Problem

7 (10%). Let $u$ and $v$ be the unit normal vectors of the tangent planes of the surfaces $S_1:x^2+y^2+z^2+2x-4y-4z=12$ and $S_2:4x^2+y^2+16z^2=24$ at the common point $(1,-2,1)$. Compute the inner product $u\cdot v$.