If $x$ is an element in a standard convex linear optimization set constrained by $Ax = b, x \geq 0$, then how can I prove $d$ is a feasible direction only if $Ad=0$ and $di \geq 0$ for every $i$ where $xi=0$?
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[EDITED after OP's clarification that $i$ is a subscript.]
If I get the idea of feasible direction right then it means that $x+\lambda d$ should be in the constraint set for some $\lambda>0$.