Inner product of vectors in LaTeX

Symbol Command
$u \cdot v$ u \cdot v
$\bm{u} \cdot \bm{v}$ \bm{u} \cdot \bm{v}
Symbol Command
$\langle u,v \rangle$ \langle u,v \rangle
$\langle u,\ v \rangle$ \langle u,\ v \rangle
$\langle u,\,v \rangle$ \langle u,\,v \rangle
$\langle \bm{u},\bm{v} \rangle$ \langle \bm{u},\bm{v} \rangle
$\langle \frac{\bm{u}}{2},\bm{v} \rangle$ \langle \frac{\bm{u}}{2},\bm{v} \rangle
$\langle \bm{\frac{u}{2}},\bm{v} \rangle$ \langle \bm{\frac{u}{2}},\bm{v} \rangle
\[
\langle \frac{\bm{u}}{2},\bm{v} \rangle \\~\\
\left\langle \frac{\bm{u}}{2},\bm{v} \right\rangle \\~\\
\langle \bm{\frac{u}{2}},\bm{v} \rangle \\~\\
\left\langle \bm{\frac{u}{2}},\bm{v} \right\rangle
\]

\[ \langle \frac{\bm{u}}{2},\bm{v} \rangle \\~\\ \left\langle \frac{\bm{u}}{2},\bm{v} \right\rangle \\~\\ \langle \bm{\frac{u}{2}},\bm{v} \rangle \\~\\ \left\langle \bm{\frac{u}{2}},\bm{v} \right\rangle \]

Symbol Command
$(u,v)$ (u,v)
$(u,\ v)$ (u,\ v)
$(u,\,v)$ (u,\,v)
$(\bm{u},\bm{v})$ (\bm{u},\bm{v})
$(\bm{u},\ \bm{v})$ (\bm{u},\ \bm{v})
$(\bm{u},\,\bm{v})$ (\bm{u},\,\bm{v})
$(\frac{\bm{u}}{2},\bm{v})$ (\frac{\bm{u}}{2},\bm{v})
$(\bm{\frac{v}{2}},\bm{v})$ (\bm{\frac{v}{2}},\bm{v})
\[
(\frac{\bm{u}}{2},\bm{v}) \\~\\
\left( \frac{\bm{u}}{2},\bm{v} \right) \\~\\
(\bm{\frac{u}{2}},\bm{v}) \\~\\
\left(\bm{\frac{u}{2}},\bm{v}\right)
\]

\[ (\frac{\bm{u}}{2},\bm{v}) \\~\\ \left( \frac{\bm{u}}{2},\bm{v} \right) \\~\\ (\bm{\frac{u}{2}},\bm{v}) \\~\\ \left(\bm{\frac{u}{2}},\bm{v}\right) \]

LaTeX vector

LaTeX