Sum of projections onto vectors which sum to zero.

Sum of projections onto vectors which sum to zero.

Premise: Let $V$ be some vector space over $K$ (in my case it's $d$
dimensional Euclidian space). Let $U= \{\mathbf u_i:i=1...n\}$ be a subset
of $V$ which has the additional properties $\sum_i \mathbf u_i = 0_V$ and
$|\mathbf u_i|=1_K \,\,\forall i$.
The projection operator $P : V \rightarrow V$ onto some vector $\mathbf
u_i \in V$ is defined $P(\mathbf v) = (\mathbf u_i \otimes \mathbf u_i)
\cdot \mathbf v$.
Consider the quantity $\left(\sum_i \mathbf u_i \otimes \mathbf
u_i\right)\cdot\mathbf v = \sum_i \mathbf u_i (\mathbf u_i \cdot\mathbf v)
:= P_{tot}(\mathbf v)$.
Question: Is there nice simplification for this quantity? Is it the case
that $P_{tot}(\mathbf v) = \lambda \mathbf v$ for some $\lambda \in K$?
Edit: I realised I was barking up the wrong tree and that in general
$P_{tot}(\mathbf v)\neq 0_V$