  We present our velocity measurements of 59 clumpy, metal-rich ejecta knots in
the supernova remnant (SNR) of SN 1572 (Tycho). We use our 450 ks Chandra High
Energy Transmission Grating Spectrometer observation to measure the Doppler
shift of the He-like Si K$\alpha$ line-center wavelength emitted from these
knots to find their line-of-sight (radial) velocities ($v_r$). We find $v_r$ up
to $\sim$ 5500 km s$^{-1}$, with roughly consistent speeds between blueshifted
and redshifted ejecta knots. We also measure the proper motions (PMs) for our
sample based on archival Chandra Advanced CCD Imaging Spectrometer data taken
from 2003, 2009, and 2015. We estimate PMs up to 0$"$.35 yr$^{-1}$, which
corresponds to a transverse velocity of about 5800 km s$^{-1}$ for the distance
of 3.5 kpc to Tycho. Our $v_r$ and transverse velocity measurements imply space
velocities of $\sim$ 1900 - 6000 km s$^{-1}$ for the ejecta knots in Tycho. We
estimate a new expansion center of R.A.(J2000) = 00$^h$25$^m$18$^s$.725 $\pm$
1$^s$.157 and decl.(J2000) = +64$^{\circ}$08$'$02$"$.5 $\pm$ 11$"$.2 from our
PM measurements, consistent to within $\sim$ 13$"$ of the geometric center. The
distribution of space velocities throughout the remnant suggests that the
southeast quadrant generally expands faster than the rest of the SNR. We find
that blueshifted knots are projected more in the northern shell, while
redshifted knots are more in the southern shell. The previously estimated
reverse shock position is consistent with most of our estimated ejecta
distribution, however some ejecta show deviations from the 1-D picture of the
reverse shock.
