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)