Vibrations at Absolute Zero Is matter still vibrating at absolute zero temperature?  If it isn't, wouldn't we then be able to know both its position and momentum?
 A: A system at absolute zero will still have nonzero uncertainty in position and momentum.
As an example, consider the simplest vibrating system, the one dimensional harmonic oscillator with (angular) frequency $\omega$ and mass $m$. In it's ground state (i.e. the lowest possible energy for the system) the position probability amplitude is a Gaussian ('bell curve'):
$$ \psi(x)\propto e^{\frac{-m \omega x^2}{2 \hbar}}$$. The finite extent of that curve reflects the position uncertainty: $\Delta x = \sqrt{\frac{\hbar}{2 m \omega}}$. One can also show that in this state the momentum uncertainty is $\Delta p= \sqrt{\frac{ m \omega \hbar}{2}}$, and it is a minimal uncertainty state ($\Delta p \Delta x = \hbar/2$). 
This example is easily generalized to 3D and is actually quite relevant since vibrating systems can often be approximated quite well by a simple harmonic oscillator, especially at low energies.
A: There will still be vibrations due to zero-point energy. So it's still impossible to know both its position and momentum.
A: You are thinking about this backwards.
Because of the uncertainty principle, zero-point energy exists.
Because of the existence of zero-point energy, it is impossible to cool any object all the way to absolute zero.
