Intersection between plane and line given the unit vector of the plane

Question

I want to find the value of the vector $\mathbf{r}_{f'}$ in the below image.

The image describes two adjacent volumes (cells) where the centroid of each cell is point $C$ and $F$.

The two cells share one face with centroid at $f$ and normal area vector $S_f$.

The vector connecting the two cell centroids is $CF$ which intersect with the shared face at point $f'$.

$e$ is the unit vector in the direction of $CF$

The following are known: $\mathbf{r}_{f}$, $\mathbf{S}_{f}$ (normal vector), $\mathbf{r}_{F} $ and $\mathbf{r}_{C}$

Thanks in advance.

Answer 1

The point $f’$ is the intersection of the lines $\overline{CF}$ and $\overline{ff'}$. It looks like you already know how to find an equation for the former. For the latter, you can use the normal form: $\mathbf S_f\cdot(\mathbf r-\mathbf r_f)=0$. I expect that you can take it from there.