We present the first three-dimensional measurements of the velocity of various ejecta knots in Tychos supernova remnant, known to result from a Type Ia explosion. Chandra X-ray observations over a 12-year baseline from 2003 to 2015 allow us to measure the proper motion of nearly 60 tufts of Si-rich ejecta, giving us the velocity in the plane of the sky. For the line of sight velocity, we use two different methods: a non-equilibrium ionization model fit to the strong Si and S lines in the 1.2-2.8 keV regime, and a fit consisting of a series of Gaussian lines. These methods give consistent results, allowing us to determine the red or blue shift of each of the knots. Assuming a distance of 3.5 kpc, we find total velocities that range from 2400 to 6600 km s$^{-1}$, with a mean of 4430 km s$^{-1}$. We find several regions where the ejecta knots have overtaken the forward shock. These regions have proper motions in excess of 6000 km s$^{-1}$. Some Type Ia supernova explosion models predict a velocity asymmetry in the ejecta. We find no such velocity asymmetries in Tycho, and discuss our findings in light of various explosion models, favoring those delayed detonation models with relatively vigorous and symmetrical deflagrations. Finally, we compare measurements with models of the remnants evolution that include both smooth and clumpy ejecta profiles, finding that both ejecta profiles can be accommodated by the observations.
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