For $\{B_H(t)= (B_{H,1}(t), \ldots, B_{H,d}(t))^\top,t\ge0\}$, where $\{B_{H,i}(t),t\ge 0\}, 1\le i\le d$ are mutually independent fractional Brownian motions, we obtain the exact asymptotics of $$ \mathbb P (\exists t\ge 0: A B_{H}(t) - μt >νu), \ \ \ \ u\to\infty, $$ where $A$ is a non-singular $d\times d$ matrix and $μ=(μ_1,\ldots, μ_d)^\top\in R^d$, $ν=(ν_1, \ldots, ν_d)^\top \in R^d$ are such that there exists some $1\le i\le d$ such that $μ_i>0, ν_i>0.$
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