Find the line Integral of the vector field $F = zx{\hat i} + xy {\hat j} + yz{\hat k}$ over the boundary of the
triangle with vertices $(1,1,0), (0,1,0) , (0,0,1)$
oriented anticlockwise , when viewed from the point $(2,2,2)$.
I would like to solve this question with help of Stoke's Theorem , but I am unable to calculate the unit normal vector $\hat{n}$.
Can anyone please tell me how to calculate the outward unit normal for this figure ?
Thank you
Define vertices of triangles as $\hat A$, $\hat B$, $\hat C$ and point of view $\hat D$. Vector $\hat D-\hat A$ has a component parallel to the triangle and the component perpendicular to the triangle, which is a scale normal $\vec n$.
We can calculate another normal $\hat n_0=(\hat B-\hat A)\times(\hat C-\hat A)$. If it looks in the same direction as $\hat D-\hat A$ (i.e. $(\hat D-\hat A)\cdot\hat n_0>0$), then $\hat n = \hat n_0/|\hat n_0|$. If not, then $\hat n = -\hat n_0/|\hat n_0|$.