  In this paper, we study the Nakano-positivity and dual-Nakano-positivity of
certain adjoint vector bundles associated to ample vector bundles. 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 $q\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive
vector bundle, the naturally induced Hermitian vector bundle $(S^kE\ts \det E,
S^kh\ts \det h)$ is both Nakano-positive and dual-Nakano-positive for any
$k\geq 0$.
