Tangent vector and cross product of two gradient vectors

Question

Say I need to find the equation of a tangent line at some specific point b to the curve
C: $x^2 + y^2 + z^2$ = 9,
4($x^2 + y^2$) = 5$z^2$.

So, C is the intersection of these two shapes. If I find the gradient vectors for these two shapes at that point b, why and how the result of the cross product of these two gradient vectors will give me the tangent vector to the curve C? I understand that the result of the cross product of two vectors will give me the vector that will be perpendicular to both of them.

I haven't found any proof or logical explanation of how that is possible.
Any help will be greatly appreciated!

Answer 1

The gradient vector is normal to the surface. What's the geometrical interpretation of the magnitude of gradient generally? That means that any vector in the tangent plane at point b is perpendicular to the surface. At the intersection of the two surfaces, at point b, each of them has a gradient, and a tangential plane. The intersection of the two planes is a line (if the gradients are not colinear). Since the intersection belongs to each of the tangent planes, it is perpendicular to both gradients. So now you have two vectors and you want to find a vector perpendicular to both of them. You get this by doing the cross product. https://en.wikipedia.org/wiki/Cross_product