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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.

enter image description here

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    The image is too messy I think. Could you please state the problem in words too? – 2017-01-18
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    @Pythagoricus thanks, I've added description of the image. If it still vague please let me know. – 2017-01-18
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    Isn’t this simply a matter of computing the intersection of the lines $\overline{ff'}$ and $\overline{FC}$? You have all of the information you need to construct the equations of these lines. – 2017-01-18
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    @amd Yes, it is. But it's not so obvious for me how to construct the equation for $\overline{ff'}$. Note that $\mathbf{r}_{f'}$ isn't known. Can you please elaborate a little bit? – 2017-01-18

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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.

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    This leads to a little bit complicated formula but after checking I found that it gives correct results – 2017-01-19