对任意两个非零的平面向量$\mathit{\alpha }$和$\mathit{\beta }$，定义$\mathit{\alpha }·\mathit{\beta }=\frac{\mathit{\alpha }·\mathit{\beta }}{\mathit{\beta }·\mathit{\beta }}$.若两个非零的平面向量$\stackrel{\rightharpoonup }{a}$和$\stackrel{\rightharpoonup }{b}$，满足$\stackrel{\rightharpoonup }{a}$与$\stackrel{\rightharpoonup }{b}$的夹角$\theta \in \left(\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}\right)$，且$\stackrel{\rightharpoonup }{a}·\stackrel{\rightharpoonup }{b}$和$\stackrel{\rightharpoonup }{b}·\stackrel{\rightharpoonup }{a}$都在集合$\left\{\frac{n}{2}|n\in Z\right\}$中，则$\stackrel{\rightharpoonup }{a}·\stackrel{\rightharpoonup }{b}$$=$（）

A．

$\frac{5}{2}$

B．

$\frac{3}{2}$

C．

$1$

D．

$\frac{1}{2}$