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As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact K\\\"ahler manifold $X$, $S^kE\\ts \\det E$ is both Nakano-positive and dual-Nakano-positive for any $k\\geq 0$. Moreover, $H^{n,q}(X,S^kE\\ts \\det E)=H^{q,n}(X,S^kE\\ts \\det E)=0$ for any $q\\geq 1$. 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