共50条信息
已知\(A(-1,\cos θ)\),\(B(\sin θ,1)\),若\(|\overrightarrow{OA}+\overrightarrow{OB}|=|\overrightarrow{OA}-\overrightarrow{OB}|(O\)为坐标原点\()\),则锐角\(θ=(\) \()\)
已知\({\triangle }{ABC}\)是边长为\(2\)的等边三角形,\(P\)为平面\(ABC\)内一点,则\(\overrightarrow{{PA}}{⋅}(\overrightarrow{{PB}}{+}\overrightarrow{{PC}})\)的最小值是\(({ })\)
已知\(A\),\(B\),\(C\)是半径为\(1\)的圆\(O\)上的三点,\(AB\)为圆\(O\)的直径,\(P\)为圆\(O\)内一点\((\)含圆周\()\),则\(\overrightarrow{{PA}}·\overrightarrow{{PB}}+\overrightarrow{{PB}}·\overrightarrow{{PC}}+\overrightarrow{{PC}}·\overrightarrow{{PA}}\)的取值范围为____.
如图,在平行四边形\(ABCD\)中,\(\angle BAD=\dfrac{{ }\!\!\pi\!\!{ }}{3}\),\(AB=2\),\(AD=1\),若\(M\)、\(N\)分别是边\(AD\)、\(CD\)上的点,且满足\(\dfrac{MD}{AD}=\dfrac{NC}{DC}=\lambda \),其中\(\lambda \in [0,1]\),则\(\overrightarrow{AN}\cdot \overrightarrow{BM}\)的取值范围是\((\) \()\)
已知向量\( \overrightarrow{a}=\left(0,-1,1\right) \),\( \overrightarrow{b}=\left(4,1,0\right) \),\(\left|λ \overrightarrow{a}+ \overrightarrow{b}\right|= \sqrt{29} \)且\(λ > 0 \),则\(λ= \)_______.
设\(A\)、\(B\)分别为双曲线\( \dfrac{{x}^{2}}{{a}^{2}}- \dfrac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)\)的左右顶点,双曲线的实轴长为\(4 \sqrt{3} \),焦点到渐近线的距离为\( \sqrt{3} \).
\((1)\)求双曲线的方程;
\((2)\)已知直线\(y= \dfrac{ \sqrt{3}}{3}x-2 \)与双曲线的右支交于\(M\)、\(N\)两点,且在双曲线的右支上存在点\(D\),使\( \overset{→}{OM}+ \overset{→}{ON}=t \overset{→}{OD} \),求\(t\)的值及点\(D\)的坐标.
已知向量若\(\overset{\to }{{a}}\,=\left( 1,1 \right),\overset{\to }{{b}}\,=\left( 2,x \right)\),若\(\overrightarrow{a}+\overrightarrow{b}\)与\(\overrightarrow{a}-\overrightarrow{b}\)平行,则实数\(x\)的值是( )
设数列\(\left\{ {{a}_{n}} \right\}\)满足对任意的\(n\in {{N}^{*}}\),\({{P}_{n}}\left( n,{{a}_{n}} \right)\)满足\(\overrightarrow{{{P}_{n}}{{P}_{n+1}}}=(1,2)\),且\({{a}_{1}}+{{a}_{2}}=4\),则数列\(\left\{ \dfrac{1}{{{a}_{n}}\cdot {{a}_{n+1}}} \right\}\)的前\(n\)项的和\({{S}_{n}}\)为_________.
设\(\triangle \)\(ABC\)的三个内角为 \(A\), \(B\), \(C\),向量\(m\) \(=\)\(\left( \sqrt{3}\sin A,\sin B\right), n\) \(=\)\((\cos \)\(B\)\(, \sqrt{3}\cos A), \)若\(m·n\) \(=\)\(1\) \(+\)\(\cos ( \)\(A+B\)\()\),则 \(C=\) .
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