= Solution
Take the outward surface normal as $[111]$ toward the observer and retain the supplied downward vector $\mathbf a=[\bar1\bar12]$. The opposite vector is immediately $\mathbf b=[11\bar2]$. With the diagram's orientation, the remaining arrows are
$$
\boxed{\mathbf c=[01\bar1],\quad
\mathbf d=[1\bar10],\quad
\mathbf e=[\bar101].}
$$
All have zero dot product with $[111]$. The last three are in-plane close-packed $\langle110\rangle$ directions; $\mathbf a,\mathbf b$ are opposite $\langle112\rangle$ directions perpendicular to the corresponding straight close-packed step edge.
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