1) Let F1 = x^2 - y^2 + z^2 -1 = 0 F2 = xy + xz - 2 = 0 F3 = xyz - x^2 - 6y + 6 = 0 My thought is to compute the gradients, grad F1 and grad F2. Then by taking their cross product, I can get a tangent vector v for the curve. Now, I can feel that if I can show that (v) dot (grad F3) = 0, then I am done. But I don't quite understand why. Can someone explain why this works? Also, provided that I've shown (v) dot (grad F3) = 0, can the curve and surface be simply "parallel" to each other, rather than "tangent" to each other? If yes, what should I do? If not, why not? 2) I am OK with parts (i) and (iii), but I am stuck with part (ii), can someone please guide me through the steps of solving this part? [Related concepts: Curves and surfaces, smoothness, implicit function theorem, inverse function theorem, transformations and coordinate systems] Any help is greatly appreciated!