let $S,T$ be respectively $k$-, $n$-tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,\ldots,x_{k+n}) := T(x_1,\ldots,x_k) S(x_{k+1}, \ldots, x_{k+n}) $$ (their product as real numbers, or product in any base field).
Now, what if either of $S,T$ is a $0$-tensor, i.e., a vector. How do we then define the product? I can tell the result would be a vector, i.e., $(0+0)$-tensor. Maybe we can use some duality to identify a $0$-tensor with a linear map, or we can use the inverse of the trace map? SERENITY NOW!!! (er, I mean, thanks for any help!).
$0$-tensors are just scalars, so the tensor product in this case is just scalar multiplication.