does angle between tangents of two surfcaces at the point of intersection is same as angle between their normals at that point.

Question

I have to find the value of a and b such that the curves:
$ax^2-byz=(a+2)x$ and $4x^2y+z^3=4$ are orthogonal at point (1,-1,2),
Now we know that angle between curves is the angle between their tangents at the point of intersection.so, by doing dot product between their gradient = 0 and putting the point in 1st curve, I correctly found b=1 and a = 5/2.
Now as per this question we are saying that angle between tangents at the point of intersection is the angle between normals of tangent plane at that point.

In another question i have to find the equation of the tangent line to intesection of curves $x^2 +y^2+z^2=1, x+y+z=1$ at point (1,0,0). In this case I found the normals to the curve at the given point which were
(i) 2 $i$ and
(ii) $i+j+k$ ,
and by cross product of these I got the direction of tangent vector.
Here comes the doubt,in 1st case the angle between tangents is the angle between their normals but in second case the angle between tangents is zero (since they have common tangent, which we found by cross product) but the angle between their normals (i) and (ii) is not zero.