已知平面向量\(\overrightarrow{a}=\left( 1,-3 \right),\overrightarrow{b}=\left( 4,-2 \right)\)，若\(\lambda \overrightarrow{a}-\overrightarrow{b}\)与\(\overrightarrow{a}\)垂直，则\(\lambda =\)(    )

\(\Delta ABC\)的三个内角\(A\)，\(B\)，\(C\)成等差数列， 且\((A\vec{B}+A\vec{C})\cdot B\vec{C}=0\)则\(\Delta ABC\)的 形状为\((\)  \()\)

在平面直角坐标系\(xOy\)中，已知向量\(\overrightarrow{m}=\left(\begin{matrix} \dfrac{ \sqrt{2}}{2},- \dfrac{ \sqrt{2}}{2}\end{matrix}\right)\)，\(\overrightarrow{n}=(\sin x,\cos x)\)，\(x∈\left(\begin{matrix}0, \dfrac{π}{2}\end{matrix}\right)\)．

\((1)\)若\(\overrightarrow{m}\bot \overrightarrow{n}\)，求\(\tan x\)的值；

\((2)\)若\(\overrightarrow{m}\)与\(\overrightarrow{n}\)的夹角为\( \dfrac{π}{3}\)，求\(x\)的值．

已知\(|\overrightarrow{a}|=2\)，\(|\overrightarrow{b}|=3\)，\(\overrightarrow{a}\)与\(\overrightarrow{b}\)的夹角为\(60^{\circ}\)，\(\overrightarrow{c}=5\overrightarrow{a}+3\overrightarrow{b}\)，\(\overrightarrow{d}=3\overrightarrow{a}+\)\(k\)\(\overrightarrow{b}\)，当实数\(k\)为何值时，

 \((1)\overrightarrow{c}/\!/\overrightarrow{d}\)； 

 \((2)\overrightarrow{c}⊥\overrightarrow{d}\)．

\((1)\)已知向量\( \overset{→}{a} =(1,\)\(m\)\()\)，\( \overset{→}{b} =(\)\(m\)，\(m\)\(-3)\)，若\( \overset{→}{a}⊥ \overset{→}{b} \)，则\(m\)\(= \)______．

\((2)\)已知\(\sin \)\(( \dfrac{π}{3} \) \(-α)= \dfrac{1}{3} (0 < α < \dfrac{π}{2} \) \()\)，则\(\sin \)\(( \dfrac{π}{6} \) \(+α)= \)______．

\((3)\)已知\(\triangle ABC\)是边长为\(2\)的等边三角形，则\( \overset{→}{AB}· \overset{→}{BC} = \)______．

\((4)\)设\(\triangle ABC\)的内角\(A\)，\(B\)，\(C\)，所对的边分别是\(a\)，\(b\)，\(c\)\(.\)若\(a\)\({\,\!}^{2}+\)\(b\)\({\,\!}^{2}-\)\(c\)\({\,\!}^{2}+\)\(ab\)\(=0\)，则角\(C= \)______．

在\(\triangle ABC\)中，边\(AC=1\)，\(AB=2\) 角\({A}=\dfrac{2\pi }{3}\)，过\({A}\)作\(AP⊥BC \)于\(P\)，且\(\overrightarrow{{AP}}=\lambda \overrightarrow{{AB}}+\mu \overrightarrow{{AC}}\)，则\(\lambda \mu =\) _________     ．