Given vectors $\displaystyle a,b\in \mathbb{R}^{3}$ and that $\displaystyle a\cdot c = b\cdot c$ and $\displaystyle a \times c = b\times c$ for some nonzero vector $\displaystyle c$, prove that $\displaystyle a=b$.

So I showed that if $\displaystyle a = 0$ then $\displaystyle b=0$, so now I assume $\displaystyle a,b$ are nonzero vectors. So $\displaystyle |a||c|\cos\theta = |b||c|\cos\phi$ and $\displaystyle |a||c|\sin\theta = |b||c|\sin\phi$ (where $\displaystyle \theta$ is the angle between $\displaystyle a$ and $\displaystyle c$ and $\displaystyle \phi$ is the angle between $\displaystyle b$ and $\displaystyle c$) and since all three vectors are nonzero, I can solve for one of the magnitudes and show that $\displaystyle |a| = |b|$ and then from there I get $\displaystyle cos\theta = \cos\phi$. Now, can I assume that the angles must be in the range of $\displaystyle 0\le \theta , \phi \le \pi$ and therefore that $\displaystyle \theta = \phi$. If that conclusion can be made, is it sufficient to show that if two vectors have the same magnitude and that the angles between them and the vector $\displaystyle c$ are the same, then they are the same vector? If not, how would I go about proving this? Thanks.