In the product manifold $S^2 \times T^2$ of an oriented 2-sphere and an oriented 2-torus, is the oriented intersection number $I(A, B) = \chi(A) \times \chi(B)$ where $A$ is the submanifold $S^2 \times \{p\}$ and $B$ is the submanifold $\{q\} \times T^2$? Or is $I(A, B) = 1$ or $I(A, B) = -1$ since the submanifolds $A$ and $B$ intersect at only one point $(q, p)$ in $S^2 \times T^2$? I'm not sure whether it should be $+1$ or $-1$ but maybe that depends on the choice of orientation on $S^2 \times T^2$.
I think $\chi(A) = I(A, A) = 2$ and $\chi(B) = I(B, B) = 0$ are the respective Euler characteristics of the 2-sphere and the 2-torus using Poincare-Hopf. However, I'm not really sure about how to calculate $I(A, B)$.