Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 1 / b

Write the surfaces as level sets $f_1=x^2{y}^{2}z-3x-z^2=0$ and $f_2=3x^2-y^2+2z-1=0$. Their gradients are normal vectors to their tangent planes. At the specified point they are ${\mathbf{n}}_1=(5,-4,2)$ and ${\mathbf{n}}_2=(6,4,2)$, with dot product $18$ and squared norms $45,56$. The angle between intersecting regular surfaces is the angle between their tangent planes, soBoth gradients are nonzero, ensuring that the tangent planes are defined.