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
E
is an ample vector bundle over a compact Kähler manifold
X
X
,
S
k
E
⊗
det
E
S^kE\otimes \det E
is both Nakano-positive and dual-Nakano-positive for any
k
≥
0
k\geq 0
. Moreover,
H
n
,
q
(
X
,
S
k
E
⊗
det
E
)
=
H
q
,
n
(
X
,
S
k
E
⊗
det
E
)
=
0
H^{n,q}(X,S^kE\otimes \det E)=H^{q,n}(X,S^kE\otimes \det E)=0
for any
q
≥
1
q\geq 1
. In particular, if
(
E
,
h
)
(E,h)
is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle
(
S
k
E
⊗
det
E
,
S
k
h
⊗
det
h
)
(S^kE\otimes \det E, S^kh\otimes \det h)
is both Nakano-positive and dual-Nakano-positive for any
k
≥
0
k\geq 0
.