To find the ratio in which B divides AC for $$A=(1,2,3)\\B=(3,4',7)\\C=(-3,-2,-5)$$
I used the section formula assuming the following format, where $\lambda:1$ is only the ratio of division; 
You end up with $\lambda=-3$ , which I interpreted to mean 'B lies on the other side of C at thrice the distance to C as to A'. This is quite obviously nonsense.
Trying again, I solved it by taking $\lambda:1$ to be the ratio in which B divides AC(with $\lambda$ and 1 interchanged in the diagram). Now I got $\lambda=\frac{-1}{3}$. This time, it makes sense; B lies on the other end of A. This is how it actually works out; I graphed it on GeoGebra.
Why did my first interpretation fail? I did think subsequently of interpreting it like; $$\lambda=-3= \frac 3 {-1}$$ so that $\lambda$ is effectively 3. You get the correct answer like this; B is at thrice the distance to C(in the same as initially assumed direction) as to A, and it's in the negative direction to A as compared to the initial assumption. But this doesn't seem rigouous; I had set the ratio as $\lambda:1$, and I have to interpret $\lambda$ as it is, or the 1 will be affected.
Observe that in actuality,
$$\vec{BC}=-3\vec{AB}.$$
$$$$
Therefore, in your original setup,
$$\lambda\,\vec v =-3(1\vec{v})\\ \lambda=-3;$$
whereas in your alternative setup,
$$1\,\vec v =-3(\lambda\vec{v})\\ \lambda=-\frac13.$$