Navier-Stokes Convective term divergence of tensor is a row vector

Consider the convective term from the Navier-Sokes equation (e.g. Navier–Stokes momentum equation (conservation form) from https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations):
$$
nabla cdot (rho vec u otimes vec u ).
$$
Writing the outer product for three dimensional Eucledian space:
$$
vec u otimes vec u = rho
begin{pmatrix}
uu & vu & wu
uv & vv & wv
uw & vw & ww
end{pmatrix}
$$
we will arrive at
$$
nabla cdot (rho vec u otimes vec u ) =
begin{pmatrix}
dfrac{partial}{partial x}
dfrac{partial}{partial y}
dfrac{partial}{partial z}
end{pmatrix}
cdot
begin{pmatrix}
rho uu & rho vu & rho wu
rho uv & rho vv & rho wv
rho uw & rho vw & rho ww
end{pmatrix}
=
begin{pmatrix}
dfrac{partial}{partial x} &dfrac{partial}{partial y} &dfrac{partial}{partial z}
end{pmatrix}
begin{pmatrix}
rho uu & rho vu & rho wu
rho uv & rho vv & rho wv
rho uw & rho vw & rho ww
end{pmatrix}
=
begin{pmatrix}
dfrac{partial rho uu}{partial x}
+ dfrac{partial rho uv}{partial y}
+ dfrac{partial rho uw}{partial z},
& dfrac{partial rho vu}{partial x}
+ dfrac{partial rho vv}{partial y}
+ dfrac{partial rho vw}{partial z},
& dfrac{partial rho wu}{partial x}
+ dfrac{partial rho wv}{partial y}
+ dfrac{partial rho ww}{partial z}
end{pmatrix}
$$
My confusion is that the result is a row vector (1 x 3) not a column one (3 x 1) which is the case for all other terms in the equation. Shouldn’t the convective term be
$$
left( nabla cdot (rho vec u otimes vec u ) right)^T ~?
$$
If we consider the NS equations in convective form (see Navier–Stokes momentum equation (convective form) from the link above) the convective term:
$$
vec u cdot nabla vec u
$$
is in my opinion ambiguous. We can evaluate it like this $vec u cdot (nabla vec u)$ or like this $(vec u cdot nabla) vec u$. The former delivers a row vector and the latter a column vector. Some authors use explicitly the second form (see Incompressible Navier–Stokes equations (convective form) on the Wiki page)

Why is this term incosistent (row vector) with other terms of NS equation? Doesn’t it matter and I should ignore this fact? Do I miss something?

I found several questions about the convective term (here or here) but none of these addresses my issue.