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## 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 gonna 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 dont think the vector we get by cross producting the two gradients has to be tangential to s1, and s2