Now I´m proving the Rodrigues' rotation formula, and have a question in the process of covering to the matrix form of formula. The below is the Rodrigues' rotation formula $$Rv ⃗= \cos(θ) v ⃗+(1-\cos(θ) )(u ̂u ⃗^T)\cdot v ⃗+sin(θ) [u ̂]_××v ⃗$$ Where $R$ is a rotation matrix and $v ⃗ = [v1,v2,v3]$，$u ̂ =[u1,u2,u3]$ is a unit column vector and $$[u ̂]_×= \left \{ \begin{matrix} 0&-u3&u2\\ u3&0&-u1\\ -u2&u1&0\\ \end{matrix} \right\} $$ is a skew-symmetric matrix of $u ⃗ $.
And this formula converts to matrix form include below :
$$R=\cos(θ) I_{3×3} +(1-\cos(θ))
\left \{
\begin{matrix}
u1 \\
u2 \\
u3\\
\end{matrix}
\right \}
\left \{
\begin{matrix}
u1&u2&u3\\
\end{matrix}
\right \}
+\sin(θ)
\left \{
\begin{matrix}
0&-u3&u2\\
u3&0&-u1\\
-u2&u1&0\\
\end{matrix}
\right\},
$$
Where $I_{3×3}$ is the $3$ by $3$ identity matrix
So my problem is how to do in this process of convert about $v ⃗$ is dropped in the both sides of the equation, and appear the identity matrix $I_{3×3}$ beside the $cos(θ)$.
As far as I know vector division is not defined so I don't know how to convert .
The formula should be
$$R\vec v= \cos(θ) \vec v +(1-\ cos(θ) )({\hat u}\hat u^T) \vec v +\sin(θ) \hat u \times \vec v$$ Convince yourself that the right-hand side is a sum of 3 elements of $\mathbb R^3$. Note that $$\cos(θ) \vec v = \cos(\theta ) I_{3\times 3} \vec v, $$ and also you should check if you didn't already know that $$\hat u \times \vec v = [\hat u]_\times \vec v$$ where the right-hand side is matrix multiplication with a vector, and the left-hand side is vector cross product. Substitute the above equalities in, and use distributivity of matrix multiplication $AD+BD+CD = (A+B+C)D$
\begin{align} R\vec v &= \cos(θ)I_{3\times 3} \vec v +(1-\ cos(θ) )({\hat u}\hat u^T) \vec v +\sin(θ) [u ̂]_\times \vec v \\ &= \Big[ \cos(θ)I_{3\times 3} +(1-\ cos(θ) )({\hat u}\hat u^T) +\sin(θ) [u ̂]_\times \Big] \vec v \end{align} Two matrices $A,B$ of the same shape are equal iff for every vector $\vec v$, $A\vec v = B\vec v$. This allows us to 'cancel' $\vec v$.