Thue has shown the existence of three types of infinite square-free words over {0,1,2} avoiding the factor 010. They respectively avoid {010,212}, {010,101}, and {010,020}. Also Dejean constructed an infinite $\left(\tfrac74^+\right)$-free ternary word. A word is $d$-directed if it does not contain both a factor of length $d$ and its mirror image. We show that there exist exponentially many $\left(\tfrac74^+\right)$-free 180-directed ternary words avoiding 010. Moreover, there does not exist an infinite $\left(\tfrac74^+\right)$-free 179-directed ternary word avoiding 010.