I have three vectors, $a$, $b$, and $c$ in $n$-dimensional space. I want to calculate $a\cdot b$. I know $\lvert a \rvert$, $\lvert b \rvert$, $\lvert c \rvert$, $a\cdot c$ and $b\cdot c$.
Is there a way to do this, preferably without using trig?
I have made some progress. If $\theta$ is the angle between $a$ and $c$, and $\phi$ is the angle between $b$ and $c$, I know that: $$a\cdot b=\lvert a\rvert\lvert b\rvert\cos(\theta-\phi)=\lvert a\rvert\lvert b\rvert\cos\theta\cos\phi+\lvert a\rvert\lvert b\rvert\sin\theta\sin\phi$$ $$=\frac{(a\cdot c)(b\cdot c)}{\lvert c\rvert^2}+\lvert a\rvert\lvert b\rvert\sin\theta\sin\phi$$
I also know that $$\lvert a\rvert^2\lvert c\rvert^2\sin^2\theta=\lvert a\rvert^2\lvert c\rvert^2-(a\cdot c)^2$$ and likewise for $b$, but this doesn't give the sign of the sines.
I think this is possible, but I'm not sure how to do it.
Edit: Okay, I realize now that this is impossible generally. Is it possible in the two-dimensional case?