在\(\Delta ABC\)中，角\(A,B,C\)的对边分别为\(a,b,c,\cos C=\dfrac{3}{10}\)．

\((1)\)若\(\overrightarrow{CA}\bullet \overrightarrow{CB}=\dfrac{9}{2}\)，求\(\Delta ABC\)的面积；

\((2)\)设向量\( \overset{⇀}{x}=(2\sin ⁡B,− \sqrt{3}), \overset{⇀}{y}=(\cos ⁡2B,1−2{\sin }^{2} \dfrac{B}{2}) \)，且\( \overset{⇀}{x}/\!/ \overset{⇀}{y} \)，求角\(B\)的值．

已知\(\left| \overrightarrow{OA} \right|=1\)，\(\left| \overrightarrow{OB} \right|=\sqrt{3}\)，向量\(\overrightarrow{OA}\)，\(\overrightarrow{OB}\)的夹角为\({{90}^{\circ }}\)，点\(C\)   在\(AB\)上，且\(\angle AOC={{30}^{\circ }}.\)设\(\overrightarrow{OC}=m\overrightarrow{OA}+n\overrightarrow{OB}(m,n\in R)\)，求\(\dfrac{m}{n}\)的值．

已知\(α∈(0, \dfrac{π}{4}),β∈( \dfrac{π}{4}, \dfrac{π}{2}), \overrightarrow{a}=(2{\cos }^{2}α,2\sin α\cos α) \)，\(\overrightarrow{b}=(1+\sin β\cos β,1-2{\sin }^{2}β), \overrightarrow{c}=(1,0)· < \overrightarrow{a}, \overrightarrow{c} > ={θ}_{1}, < \overrightarrow{b}, \overrightarrow{c} > ={θ}_{2} ..\)

\((1)\)若\({θ}_{2}= \dfrac{π}{6} \)，求角\(β\)； 

\((2)\)若\({θ}_{2}-{θ}_{1}= \dfrac{π}{6} \)，求\(\sin (β-α)\)．

\((2)\)设\(z\)，\(\bar{z}\)，\(3\bar{z}\)在复平面上对应的点分别为\(A,B,C\)，判断\(\Delta ABC\)的形状，并求\(\Delta ABC\)的面积．

已知向量\(\overrightarrow{a}=\left( 4,3 \right),\overrightarrow{b}=\left( -1,2 \right)\)，求

\(⑴\)求\(\overrightarrow{a}\)与\(\overrightarrow{b}\)的夹角\(\theta \)的余弦值；

\(⑵\)若向量\(\overrightarrow{a}-\lambda \overrightarrow{b}\)与\(2\overrightarrow{a}+\overrightarrow{b}\)垂直，求\(\lambda \)的值．