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!