  The near orthgonality of certain $k$-vectors involving the Ramanujan sums
were studied by E. Alkan in [J. Number Theory, 140:147--168 (2014)]. Here we
undertake the study of similar vectors involving a generalization of the
Ramanujan sums defined by E. Cohen in [Duke Math. J., 16(2):85--90 (1949)]. We
also prove that the weighted average $\frac{1}{k^{s(r+1)}}\sum
\limits_{j=1}^{k^s}j^rc_k^{(s)}(j)$ remains positve for all $r\geq 1$. Further,
we give a lower bound for $\max\limits_{N}\left|\sum
\limits_{j=1}^{N^s}c_k^{(s)}(j) \right|$.
