Evaluating an Equation Using Einstein Summation Notation Problem
As part of a proof for the commutation relation of a vector operator and the angular momentum operator, I need the evaluate the expression
$$(R_\omega)_{ij}V_j = \big([1-\cos(\omega)]\hat{\omega}_i\hat{\omega}_j + \cos(\omega)\delta_{ij} + \sin(\omega)\varepsilon_{ijk}\hat{\omega}_k\big) V_j$$
Here, $(R_\omega)_{ij}$ is the representation for the rotation generator I was given and advised to use, and $V_j$ is the $j$-component of said vector operator. (Eventually, I'm trying to show that $[L_i,V_j]=i\hbar\varepsilon_{ijk}V_k$.)
Attempt
Now, on the LHS, $j$-index is repeated, so I should sum over $j=i,j,k$. I'm still clunky with this convention, so I computed the sum term-wise, (with $\hat{\omega}_i\hat{\omega}_j=\delta_{ij}$ assuming orthonormal unit vectors)
$$ (R_\omega)_{ij}V_j = \big(1-\sin(\omega)\hat{\omega}_k\big)V_j $$
Is this correct? I have reservations, because (a) why was I given $\hat{\omega}_i\hat{\omega}_j$ instead of another $\delta$-function? and 
(b) this result leaves me with a scalar minus a vector, which seems erroneous to me.
I feel like my execution of the Einstein Summation notation is flawed. I'm familiar with the mechanics of it, but again, I'm clunky with implementation. For instance, am I suppose to sum over $j$ on the LHS and then $i,j$ on the RHS?
 A: Hint :
Your notation is a little confusing. Here $\;\boldsymbol{\hat{\omega}}\;$ is a unit vector:
\begin{equation}
\boldsymbol{\hat{\omega}}\boldsymbol{=}\left(\hat{\omega}_1,\hat{\omega}_2,\hat{\omega}_3\right)\,,\quad \Vert\boldsymbol{\hat{\omega}}\Vert^2\boldsymbol{=}\hat{\omega}^2_1\boldsymbol{+}\hat{\omega}^2_2\boldsymbol{+}\hat{\omega}^2_3\boldsymbol{=}1 
\tag{01}\label{01}
\end{equation}
The rotation is around this unit vector through an angle $\;\omega$. This is inconvenient notation since anyone would think that $\;\omega\;$ is the magnitude of $\;\boldsymbol{\hat{\omega}}$. So let replace $\;\boldsymbol{\hat{\omega}}\;$ by the unit vector $\;\boldsymbol{n}\;$
\begin{equation}
\boldsymbol{n}\boldsymbol{=}\left(n_1,n_2,n_3\right)\,,\quad \Vert\boldsymbol{n}\Vert^2\boldsymbol{=}n^2_1\boldsymbol{+}n^2_2\boldsymbol{+}n^2_3\boldsymbol{=}1 
\tag{02}\label{02}
\end{equation}
and angle $\;\omega\;$ by the angle $\;\theta$. 
Then your equation is
\begin{equation}
(R_\theta)_{ij}V_j = \bigl[\left(1\boldsymbol{-}\cos\theta\right)n_i n_j \boldsymbol{+} \delta_{ij}\cos\theta\boldsymbol{+}\sin\theta\varepsilon_{ijk}n_k\bigr] V_j
\tag{03}\label{03}
\end{equation}
in vector form(1)
\begin{equation}
R_\theta\boldsymbol{V}\boldsymbol{=}\cos\theta\cdot\boldsymbol{V}\boldsymbol{+}\left(1\boldsymbol{-}\cos\theta\right)\left(\boldsymbol{n}\boldsymbol{\cdot}\boldsymbol{V}\right)\boldsymbol{n}\boldsymbol{+} \sin\theta\left(\boldsymbol{n}\boldsymbol{\times}\boldsymbol{V}\right)
\tag{04}\label{04}
\end{equation}
Note that the term $\;\left(\boldsymbol{n}\boldsymbol{\cdot}\boldsymbol{V}\right)\boldsymbol{n}\;$ is the projection of $\;\boldsymbol{V}\;$ on the axis $\;\boldsymbol{n}\;$
\begin{equation}
\rm P_{\boldsymbol{n}}\boldsymbol{V}\boldsymbol{=}\left(\boldsymbol{n}\boldsymbol{\cdot}\boldsymbol{V}\right)\boldsymbol{n}=\left(\boldsymbol{n}\boldsymbol{n}^{\boldsymbol{\top}}\right)\boldsymbol{V}=
\begin{pmatrix}
\begin{bmatrix}
n_1\\
n_2\\
n_3
\end{bmatrix}
\begin{bmatrix}
n_1 & n_2 & n_3 \vphantom{\dfrac{a}{b}}
\end{bmatrix}
\end{pmatrix}
\begin{bmatrix}
V_1\\
V_2\\
V_3
\end{bmatrix}
\tag{05}\label{05}
\end{equation}
that is
\begin{equation}
\rm P_{\boldsymbol{n}}\boldsymbol{=}\boldsymbol{n}\boldsymbol{n}^{\boldsymbol{\top}}=
\begin{bmatrix}
n_1\\
n_2\\
n_3
\end{bmatrix}
\begin{bmatrix}
n_1 & n_2 & n_3 \vphantom{\dfrac{a}{b}}
\end{bmatrix}
=
\begin{bmatrix}
n^2_1 & n_1n_2 & n_1n_3\\
n_2n_1 & n^2_2 & n_2n_3\\
n_3n_1 & n_3n_2 & n^2_3
\end{bmatrix}
\tag{06}\label{06} 
\end{equation}
or
\begin{equation}
\left(\rm P_{\boldsymbol{n}}\right)_{ij}\boldsymbol{=}n_in_j
\tag{07}\label{07}
\end{equation}
$===================================================$
(1)
See equation (07) in my answer there :Rotation of a vector

