向量经典试题及答案

向量经典试题及答案

一、单项选择题（每题2分，共10题）1.向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow{b}=(3,x)\)，若\(\overrightarrow{a}\parallel\overrightarrow{b}\)，则\(x=\)（）A.2B.4C.6D.82.已知向量\(\overrightarrow{a}=(-1,1)\)，\(\overrightarrow{b}=(1,0)\)，则\(\vert\overrightarrow{a}+\overrightarrow{b}\vert=\)（）A.\(\sqrt{2}\)B.\(\sqrt{3}\)C.2D.33.向量\(\overrightarrow{a}=(2,-3)\)，\(\overrightarrow{b}=(m,6)\)，若\(\overrightarrow{a}\perp\overrightarrow{b}\)，则\(m=\)（）A.9B.-9C.4D.-44.已知\(\vert\overrightarrow{a}\vert=3\)，\(\vert\overrightarrow{b}\vert=4\)，且\(\overrightarrow{a}\)与\(\overrightarrow{b}\)夹角为\(60^{\circ}\)，则\(\overrightarrow{a}\cdot\overrightarrow{b}=\)（）A.6B.\(6\sqrt{3}\)C.12D.\(12\sqrt{3}\)5.若向量\(\overrightarrow{AB}=(1,2)\)，\(\overrightarrow{AC}=(3,4)\)，则\(\overrightarrow{BC}=\)（）A.\((2,2)\)B.\((-2,-2)\)C.\((4,6)\)D.\((-4,-6)\)6.向量\(\overrightarrow{a}=(1,1)\)，将\(\overrightarrow{a}\)按向量\(\overrightarrow{b}=(2,0)\)平移后得到的向量是（）A.\((3,1)\)B.\((1,1)\)C.\((-1,1)\)D.\((1,-1)\)7.已知\(\overrightarrow{a}=(x,1)\)，\(\overrightarrow{b}=(3,x)\)，若\(\overrightarrow{a}\)与\(\overrightarrow{b}\)方向相反，则\(x=\)（）A.\(\sqrt{3}\)B.\(-\sqrt{3}\)C.\(\pm\sqrt{3}\)D.08.向量\(\overrightarrow{a}=(2,-1)\)，\(\overrightarrow{b}=(-1,2)\)，则\((2\overrightarrow{a}+\overrightarrow{b})\cdot\overrightarrow{a}=\)（）A.6B.5C.4D.39.若\(\overrightarrow{a}\)，\(\overrightarrow{b}\)是非零向量，且\(\vert\overrightarrow{a}+\overrightarrow{b}\vert=\vert\overrightarrow{a}-\overrightarrow{b}\vert\)，则\(\overrightarrow{a}\)与\(\overrightarrow{b}\)的夹角为（）A.\(30^{\circ}\)B.\(45^{\circ}\)C.\(60^{\circ}\)D.\(90^{\circ}\)10.已知向量\(\overrightarrow{a}=(1,n)\)，\(\overrightarrow{b}=(-1,n)\)，若\(2\overrightarrow{a}-\overrightarrow{b}\)与\(\overrightarrow{b}\)垂直，则\(n^2=\)（）A.1B.2C.3D.4二、多项选择题（每题2分，共10题）1.下列关于向量的说法正确的是（）A.零向量与任意向量平行B.向量\(\overrightarrow{a}\)与\(-\overrightarrow{a}\)长度相等C.若\(\overrightarrow{a}\parallel\overrightarrow{b}\)，\(\overrightarrow{b}\parallel\overrightarrow{c}\)，则\(\overrightarrow{a}\parallel\overrightarrow{c}\)D.两个相等向量起点、终点一定相同2.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow{b}=(-2,1)\)，则（）A.\(\overrightarrow{a}\perp\overrightarrow{b}\)B.\(\vert\overrightarrow{a}\vert=\vert\overrightarrow{b}\vert\)C.\(\overrightarrow{a}+\overrightarrow{b}=(-1,3)\)D.\(\overrightarrow{a}\)与\(\overrightarrow{b}\)夹角为\(90^{\circ}\)3.以下向量运算正确的是（）A.\(\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a}\)B.\((\overrightarrow{a}+\overrightarrow{b})+\overrightarrow{c}=\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c})\)C.\(\lambda(\overrightarrow{a}+\overrightarrow{b})=\lambda\overrightarrow{a}+\lambda\overrightarrow{b}\)D.\((\lambda+\mu)\overrightarrow{a}=\lambda\overrightarrow{a}+\mu\overrightarrow{a}\)4.若向量\(\overrightarrow{a}=(x_1,y_1)\)，\(\overrightarrow{b}=(x_2,y_2)\)，则下列能使\(\overrightarrow{a}\parallel\overrightarrow{b}\)的条件是（）A.\(x_1y_2-x_2y_1=0\)B.\(x_1y_1-x_2y_2=0\)C.\(\frac{x_1}{x_2}=\frac{y_1}{y_2}\)（\(x_2\neq0\)，\(y_2\neq0\)）D.存在非零实数\(\lambda\)，使得\(\overrightarrow{a}=\lambda\overrightarrow{b}\)5.已知向量\(\overrightarrow{a}=(3,-4)\)，\(\overrightarrow{b}=(6,m)\)，若\(\overrightarrow{a}\)与\(\overrightarrow{b}\)共线，则下列说法正确的是（）A.\(m=-8\)B.\(\vert\overrightarrow{b}\vert=10\)C.\(\overrightarrow{a}\cdot\overrightarrow{b}=30\)D.\(2\overrightarrow{a}-\overrightarrow{b}=(0,-4)\)6.向量\(\overrightarrow{a}=(-1,2)\)，\(\overrightarrow{b}=(3,-1)\)，则（）A.\(\overrightarrow{a}\cdot\overrightarrow{b}=-5\)B.\(\vert\overrightarrow{a}+\overrightarrow{b}\vert=\sqrt{13}\)C.\(\overrightarrow{a}\)在\(\overrightarrow{b}\)上的投影向量为\(-\frac{5}{10}\overrightarrow{b}\)D.与\(\overrightarrow{a}\)同向的单位向量为\((-\frac{\sqrt{5}}{5},\frac{2\sqrt{5}}{5})\)7.对于向量\(\overrightarrow{a}\)，\(\overrightarrow{b}\)，\(\overrightarrow{c}\)，下列命题正确的是（）A.若\(\overrightarrow{a}\cdot\overrightarrow{b}=\overrightarrow{a}\cdot\overrightarrow{c}\)，则\(\overrightarrow{b}=\overrightarrow{c}\)B.\((\overrightarrow{a}\cdot\overrightarrow{b})\overrightarrow{c}=\overrightarrow{a}(\overrightarrow{b}\cdot\overrightarrow{c})\)C.\(\vert\overrightarrow{a}\cdot\overrightarrow{b}\vert\leqslant\vert\overrightarrow{a}\vert\vert\overrightarrow{b}\vert\)D.\(\vert\overrightarrow{a}+\overrightarrow{b}\vert^2=(\overrightarrow{a}+\overrightarrow{b})\cdot(\overrightarrow{a}+\overrightarrow{b})\)8.已知\(\overrightarrow{a}\)，\(\overrightarrow{b}\)是平面内两个不共线向量，\(\overrightarrow{AB}=\overrightarrow{a}+2\overrightarrow{b}\)，\(\overrightarrow{BC}=-5\overrightarrow{a}+6\overrightarrow{b}\)，\(\overrightarrow{CD}=7\overrightarrow{a}-2\overrightarrow{b}\)，则（）A.\(A\)，\(B\)，\(D\)三点共线B.\(\overrightarrow{BD}=2\overrightarrow{a}+4\overrightarrow{b}\)C.\(\overrightarrow{AC}=-4\overrightarrow{a}+8\overrightarrow{b}\)D.\(\overrightarrow{AD}=3\overrightarrow{a}+6\overrightarrow{b}\)9.设向量\(\overrightarrow{a}=(1,-1)\)，\(\overrightarrow{b}=(m+1,2m-4)\)，若\(\overrightarrow{a}\perp\overrightarrow{b}\)，则\(m\)的值可以是（）A.1B.2C.3D.410.已知向量\(\overrightarrow{a}=(2,1)\)，\(\overrightarrow{b}=(-1,k)\)，若\(\overrightarrow{a}\cdot\overrightarrow{b}=0\)，则（）A.\(k=2\)B.\(\vert\overrightarrow{b}\vert=\sqrt{5}\)C.\(\overrightarrow{a}\)与\(\overrightarrow{b}\)夹角为\(90^{\circ}\)D.以\(\overrightarrow{a}\)，\(\overrightarrow{b}\)为邻边的平行四边形面积为\(5\)三、判断题（每题2分，共10题）1.向量可以比较大小。（）2.若\(\overrightarrow{a}\cdot\overrightarrow{b}=0\)，则\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow{b}=\overrightarrow{0}\)。（）3.单位向量都相等。（）4.向量\(\overrightarrow{a}\)与向量\(\overrightarrow{b}\)的夹角范围是\([0,\pi]\)。（）5.若\(\overrightarrow{a}=(x_1,y_1)\)，\(\overrightarrow{b}=(x_2,y_2)\)，则\(\overrightarrow{a}-\overrightarrow{b}=(x_1-x_2,y_1-y_2)\)。（）6.若\(\vert\overrightarrow{a}\vert=\vert\overrightarrow{b}\vert\)，则\(\overrightarrow{a}=\overrightarrow{b}\)。（）7.零向量没有方向。（）8.向量\(\overrightarrow{a}\)在向量\(\overrightarrow{b}\)上的投影是一个向量。（）9.若\(\overrightarrow{a}\)，\(\overrightarrow{b}\)共线，则存在唯一实数\(\lambda\)，使得\(\overrightarrow{a}=\lambda\overrightarrow{b}\)。（）10.\((\overrightarrow{a}+\overrightarrow{b})^2=\overrightarrow{a}^2+2\overrightarrow{a}\cdot\overrightarrow{b}+\overrightarrow{b}^2\)。（）四、简答题（每题5分，共4题）1.已知向量\(\overrightarrow{a}=(1,-2)\)，\(\overrightarrow{b}=(-3,4)\)，求\(\overrightarrow{a}+\overrightarrow{b}\)，\(\overrightarrow{a}-\overrightarrow{b}\)的坐标。答案：\(\overrightarrow{a}+\overrightarrow{b}=(1-3,-2+4)=(-2,2)\)；\(\overrightarrow{a}-\overrightarrow{b}=(1-(-3),-2-4)=(4,-6)\)。2.已知\(\vert\overrightarrow{a}\vert=3\)，\(\vert\overrightarrow{b}\vert=4\)，\(\overrightarrow{a}\)与\(\overrightarrow{b}\)夹角为\(120^{\circ}\)，求\(\overrightarrow{a}\cdot\overrightarrow{b}\)。答案：根据向量数量积公式\(\overrightarrow{a}\cdot\overrightarrow{b}=\vert\overrightarrow{a}\vert\vert\overrightarrow{b}\vert\cos\theta\)，这里\(\theta=120^{\circ}\)，\(\cos120^{\circ}=-\frac{1}{2}\)，所以\(\overrightarrow{a}\cdot\overrightarrow{b}=3\times4\times(-\frac{1}{2})=-6\)。3.已知向量\(\overrightarrow{a}=(2,3)\)，\(\overrightarrow{b}=(-1,2)\)，若\(m\overrightarrow{a}+\overrightarrow{b}\)与\(\overrightarrow{a}-2\overrightarrow{b}\)平行，求\(m\)的值。答案：\(m\overrightarrow{a}+\overrightarrow{b}=(2m-1,3m+2)\)，\(\overrightarrow{a}-2\overrightarrow{b}=(4,-1)\)。因为两向量平行，则\(-(2m-1)-4(3m+2)=0\)，解得\(m=-\frac{1}{2}\)。4.已知向量\(\overrightarrow{a}\)，\(\overrightarrow{b}\)满足\(\vert\overrightarrow{a}\vert=1\)，\(\vert\overrightarrow{b}\vert=2\)，\(\vert\overrightarrow{a}-\overrightarrow{b}\vert=\sqrt{7}\)，求\(\overrightarrow{a}\)与\(\overrightarrow{b}\)的夹角。答案：对\(\vert\overrightarrow{a}-\overrightarrow{b}\vert=\sqrt{7}\)两边平方得\(\vert\overrightarrow{a}\vert^2-2