Very close @EddieRocks.
Your x- and z- values look correct but your y- value is slightly off.
It could just be a typo though, so do the final addition step again and see if the mistake is just cropping up at the end there.
I make the answer:
(113/147, 94/147, -2/147)
or
Here are the full calcs (using fractions so I can do it by hand):
r = 2(-(1/3, -2/3, -2/3) • (2/7, 6/7, 3/7)) x (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3)
r = 2(16/21) x (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3)
r = 32/21 (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3)
r = (64/147, 192/147, 96/147) + (1/3, -2/3, -2/3)
r = (64/147, 192/147, 96/147) + (49/147, -98/147, -98/147)
r = (113/147, 94/147, -2/147)
r= (0.769, 0.639, -0.014)