高二数学向量试题及答案

高二数学向量试题及答案

一、单项选择题(每题2分，共20分)

1.向量\(\\e。匕}=[1,2)\),\(\vec{b}=(2,x)\),若

\(\vec{a}\parallel\vec{b}\),则、(x=(\)\)

A.1B.2C.3D.4

2.已知\(\vert\vec{a}\vert=3\),\(\vert\vec{b}\vert=4\),

且'(\vec{a}\)与\(\vec{b}\)夹角为\(60-{\circ}\),则

\(\vec{a}\cdot\vec{b}=(\)\)

A.6B.\(6\sqrt{3}\)C.12D.\(12\sqrt{3}\)

3.若向量\(\vec{a}=(-1,1)\),\(\vec{b}=(3,m)\),

\(\vec{a}\perp\vec{b}\),则\如=(\)\)

A.3B.—3C.1D.-1

4.已知'(A(1,2八),\(B(3,4)\),则\(\overrightarrow{AB}=(\)\)

A.\((2,2)\)B.\((-2,-2)\)C.\((1,1)\)D.\((-1,-

D\)

5.向量\(\vec{a}=[2,-3)\),\(\vec(b)=(-1,1)\),则

\(3\vec{a}-2\vec{b}=(\)\)

A.\((8,-11)\)B.\((4,-5)\)C.\((-8,H)\)D.\((-

4,5)\)

6.<\(\vec{e_l)\),\(\vec{e_2}\)是单位向量，且

\(\vec{e_l}\cdot\vec{e_2)=-\frac{1}{2}\),则、(\vec{e_l}\)与

\(\vec{e_2}\)夹角为\((\)\)

A.\(30^{\circ}\)B.\(60^{\circ}\)C.\(120~{\circ}\)D.

\(150^{\circ}\)

7.已知向量\(\vec{a}=(x,D\),\(\vec{b}=(4,x)\),且\(\vec{a}\)

与'(\vec{b}\)方向相同，则\(x=(\)\)

A.2B.-2C.\(\pm2\)D.0

8.设向量\(\vec{a}\),\(\vec{b}\)满足\(\vert\vec{a}\vert=

1\),\(\vert\vec{b}\vert=2\),

\(\vert\vec{a}+\vec{b}\vert=\sqrt{7}\),则\(\vec{a}\)与

\(\vec{b}\)夹角为\((\)\)

A.\(30，'{\circ}\)B.\(45^{\circ}\)C.\(60^{\circ}\)I).

\(90^{\circ}\)

9.已知'(\vec{a}=(l,-2)\),\(\vec{b}=(m,4)\),且\("匕}\)

在\(\vec{b}\)上投影为\(-\sqrt{5}\),则\(m=(\)\)

A.2B.-2C.4D.-4

10.若向量\(\vec{a}=(l,D\),\(\vec{b}=(l,-1)\),

\(\vec{c}=(-1,2)\),则\(\vec{c}=(\)\)

A.\(-\frac{1}{2}\vcc{a}+\frac{3}{2}\vcc{b}\)B.

\(\frac{l}{2}\vec{a}-\frac{3}{2}\vec{b}\)

C.\(\frac{3}{2}\vec{a}-\frac{l}{2}\vec{b}\)D.\(-

\frac{3}{2}\vec{a}+\frac{l}{2}\vec{b}\)

二、多项选择题（每题2分，共20分）

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.若'(\vec{a}\parallel\vec{b}\),\(\vec{b}\parallel\vec{c}\),

则\(\vec{a}\parallel\vec{c}\)

C.单位向量都相等

I).向量的模是非负实数

2.已知向量\(\vec{a}=(l,2)\),\(\vec{b}=(-2,m)\),若

\(\vec{a}\parallel\vec{b}\),则下列说法正确的是()

A.\(m=-4\)

B.\(\vec{a}\cdot\vcc{b}=-10\)

C.\(\vert\vec{a}+\vec{b}\vert=\sqrt{5}\)

D.\(\vec{a}\)与\(\vec{b}\)方向相反

3.若向量\(\丫@(:{0}二&」,丫」)\),\(\vec{b}=(x_2,y_2)\),则下列

能使'(\vec{a}\perp\vec{b}\)的条件有()

A.\(x_lx_2+y_ly_2=0\)

B.\(\vec{a}\cdot\vec{b}=0\)

C.\(\vert\vec{a}+\vec{b)\vert=\vert\vec{a}-\vec{b}\vert\)

D.\(\vec{a}\)与\(\vec(b)\)夹角为\(9(T{\circ}\)

4.已知'(A(l,0八),\(B(3,1)\),\(C(2,0)\),则()

A.\(\overrightarrow{AB}=(2,1)\)

B.\(\overrightarrow{AC}=(1,0)\)

C.\(\overrightarrow{AB}\cdot\overrightarrow{AC}=2\)

I).\(\vert\overrightarrow{AB}\vert=\sqrt{5}\)

5.设向量\(\vec{a}=(l,-1)\),\(\vec{b}=(-1,2)\),则()

A.\(\vec{a}\cdot\vec{b}=-3\)

B.\(2\vec{a}+\vec{b}=(l,O)\)

C.\(\vert\vec{a}-\vec{b}\vert=\sqrt{13}\)

D.\(\vec{a}\)与\(\vec{b}\)夹角余弦值为'(-

\frac{3\sqrt{10}}{10}\)

6.下列向量中，与向量\(\vcc{a}=(l,\sqrt{3})\)共线的向量有1)

A.\((\frac{1}{2},\frac{\sqrt⑶}⑵)\)

B.\((-\frac{l}{2},-\frac{\sqrt{3}}{2})\)

C.\((\sqrt{3},1)\)

D.\((-\sqrt{3},-1)\)

7.已知向量\(\vec{a}\),\(\vec{b}\)满足\(\vert\vec{a}\vert=

2\),\(\vert\vec(b)\vert=3\),\(\vec{a}\cdot\vec{b}=-3\),

贝以)

A.\(\vec{a}\)与、(\vec{b}\)夹角为\(12(T{\circ}\)

B.\(\vert\vec{a}+\vec{b)\vert=\sqrt{7}\)

C.\(\vert\vec{a}-\vec{b)\vert=\sqrt{19}\)

D.\((\vec{a}+2\vec{b})\cdot(\vec{a}-\vec{b})=-l\)

8.对于非零向量\(\vec{a}\),\(\vec{b}\),下列说法正确的是()

A.若\(\vert\vec{a}+\vec{b}\vert=\vert\vec{a}\vert-

\vert\vec{b}\vert\),则\(\vec{a}\)与\(\vec{b}\)共线且方向相反

B.若\(\vec{a}\cdot\vec{b}=0\),则\(\vec{a}\perp\vec{b}\)

C.若\(\丫6(:{@}《(101\丫6(^{13}\810\),则\(\vec{a}\)与\(\vec{b}\)

夹角为锐角

I).

\(\vec{a}\cdot\vec{b}=\vert\vec{a}\vert\vert\vec{b}\vert\cos\

langle\vec{a},\vec{b}\rangle\)

9.已知向量\(\vec{a}=(3,-4)\),向量\(\vec{b}\)满足

\(\vec{b}\parallel\vec{a}\),且\(\vert\vec{b}\vert=10\),则

\(\vec{b}=(\)\)

A.\((6,-8)\)

B.\((-6,8)\)

C.\((-8,6)\)

D.\((8,-6)\)

10.设向量\(\vec{a}=(x,y)\),\(\vec{b}=(l,2)\),若\(\vec量}\)

在\(\vec{b}\)上投影为\(\sqrt{5}\),则()

A.\(x+2y=5\)

B.\(x-2y=5\)

C.满足条件的'(\vec{a}\)有无数个

D.\(\vec{a}\cdot\vec{b}=5\)

三、判断题(每题2分，共20分)

1.向量'(\vec{a}\)与向量\(-\vec{a}\)的模相等。()

2.若\(\vec{a}\cdot\vec{b}=0\),则\(\vec{a}=0\)或

\(\vec{b}=0\)o()

3.两个单位向量的和向量一定是单位向量。（）

4.若\(\vec{a}\)与\(\vec{b}\)是共线向量，则\(A\)、\(B\)、

\(C\)、\(D\)四点共线。()

5.向量'(\vec{a}=(l,2)\)与向量\(\vec{b}=(2,4)\)夹角为

\(Oj\circ}\)o()

6.若'(\vert\vec{a}\vert=\vert\vec{b}\vert\),则

\(\vec{a}=\vec{b}\)o()

7.向量'(\vec{a}\)在向量\(\vec{b}\)上投影一定是非负实数。()

8.若\(\vec{a}\),\(\vec{b}\),\(\vec{c}\)为非零向量，

\((\vec{a}\cdot\vec{b})\vec{c}=\vec{a}(\vec{b}\cdot\vec{c})\)。

()

9.若\(\丫00匕}\)与\(\vec{b}\)夹角为\(\theta\),则

\(\cos\theta=\frac{\vec{a}\cdot\vec{b}}{\vert\vec{a}\vert\ver

t\vec{b}\vert}\)o()

10.已知'(\vec{a}二(x_l,y_l)\),\(\vec{b}=(x_2,y_2)\),若

\(\vec{a}\parallel\vec{b}\),则\(x_ly_2_x_2y_l=0\)0()

四、简答题(每题5分，共20分)

1.已知向量\(\vec{a}=(l,-2)\),\(\vec{b}=(-3,4)\),求

\(\vec{a}+\vec{b}\)的模。

答案：\(\vec{a}+\vec{b}=(l-3,-2+4)=(-2,2)\),

\(\vert\vec{a}+\vec{b}\vert=\sqrt{(-2)"2+2"2}=2\sqrt{2}\)o

2.已知向量\(\vec{a}=(2,3)\),\(\vec{b}=(-1,m)\),且

\(\vec{a}\perp\vec{b}\),求\(m\)的值。

答案：因为\(\丫6/@}\「6中\丫2。{1)}\),所以

\(\vec{a}\cdot\vec{b}=0\),即\(2\times(T)+3m=0\),解得

\(ni=\fraM2N3}\)。

3.已知'(\vert\vec{a}\vert=3\),\(\vert\vec{b}\vert=4\),

\(\vec{a}\)与\(\vec{b}\)夹角为\(60-{\circ}\),求

\(\vec{a}\cdot\vec{b}\)。

答案：根据向量数量积公式

\(\vec{a}\cdot\vec{b}=\vert\vec{a}\vert\vert\vec(b)\vert\cos\

theta\),这里\(\theta=60c{\circ}\),所以

\(\vec{a}\cdot\vec{b}=3\times4\times\cos60*{\circ}=3\times4\t

imes\frac{l}{2}=6\)O

4.已知向量\(\vec{a}=(1,2八),将\(\vec{a}\)按向量

r

\(\vec{b}=(3,4)\)平移后得到向量'(\vec{a}*\),求\(\vec{a}\)o

答案:向量平移后不变，所以'(\vec平},二(1,2)\)。

五、讨论题(每题5分，共20分)

1.讨论向量共线和垂直在解析几何中的应用。

答案：在解析几何中，向量共线