Finding a tangent vector to the intersection of two surfaces

[hangainlover]

Homework Statement

The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y
intersect at a curve gamma
. Find a tangent vector to
at the point (0, 2, 4).

Homework Equations

i thought about finding gradients of the two functions and plug in the given point in the gradients and cross product the two.
I get (2,2,8) as the vector tangent to the intersection gamma.

My solution was saying " If we produce two normal vectors n1 ? S1 and n2 ? S2, any
vector perpendicular to n1 and n2 (specifically n1 × n2) would be tangent to
both S1 and S2 and thus tangent to their common intersection "
.
it didnt make sense to me.
Cause if the vector is tangent to s2, x2 + y2 = 2x + 2y, then the vector should have zero for z component. the intersection is going to be some kinda curve , so i think there will be infinite number of vectors tangential to the intersectino between s1 and s2. and the cross product of the two gradients is just one of them.
I don't think the vector we get by cross producting the two gradients has to be tangential to s1, and s2

The Attempt at a Solution

 

[hangainlover]

i also noted the the intersection is part of z=2x+2y
so why not just find a vector orthogonal to the function z=2x+2y
'?
can't any vector be tangential to a curve? i suppose we can just write a random vector and say it is tangential to the given intersection.
im confused

 

[hangainlover]

in general, i know you need to find two gradients and cross them to get the vector tangential to the intersection of the two.
but in this case, since the intersection is just a plane, i thought there can be infinite number of vectors tangential to the intersection.

 

[Mark44]

Homework Statement

The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y
intersect at a curve gamma
. Find a tangent vector to
at the point (0, 2, 4).

Do x2 and y2 happen to mean x² and y²? If so, at the very least write these as x^2 and y^2 to give readers the tiniest clue as to what you are talking about.