關于平面向量

,

,

,有下列命題:
①

-

=0
②|

|-|

|<|

-

|;
③

-

不與

垂直;
④非零向量

和

滿足|

|=|

|=|

-

|,則

與

-

的夾角為60°.
其中真命題的個數為( )
A.1個
B.2個
C.3個
D.4個
【答案】
分析:由于 (

•

)

表示一個與

平行的向量,而(

•

)

表示一個與

平行的向量,故①不一定成立.
當

=

時,②不成立.
根據[(

•

)

-(

•

)

]•

=0,得到(

•

)

-(

•

)

與

垂直,故③不正確.
④由非零向量

和

滿足|

|=|

|=|

-

|,可得向量

和

、

這三個向量構成一個等邊三角形,故④正確.
解答:解:由于 (

•

)

表示一個與

平行的向量,而(

•

)

表示一個與

平行的向量,而

與

的大小方向都不確定,
故①不一定成立.
當

=

時,|

|-|

|=|

-

|=0,故②不成立.
[(

•

)

-(

•

)

]•

=(

•

)•(

•

)-(

•

)•(

•

)=0,故(

•

)

-(

•

)

與

垂直,
故③不正確.
④非零向量

和

滿足|

|=|

|=|

-

|,∴向量

和

、

這三個向量構成一個等邊三角形,
則

與

+

的夾角為30°,故④正確.
故選A.
點評:本題考查兩個向量的數量積公式的應用,兩個向量的加減法的法則,以及其幾何意義,正確利用兩個向量運算的
幾何意義,是解題的難點.
練習冊系列答案
相關習題
科目:高中數學
來源:2012-2013學年安徽省馬鞍山市高一(上)期末數學試卷(解析版)
題型:填空題
關于平面向量

,

,

,有下列幾個命題:
①若

,則

或

;
②若

與

均為單位向量,它們的夾角為60°,則

;
③若非零向量

,

,

滿足

,

,則

與

的夾角為120°;
④若

,

,則

在

方向上的投影是-1.
其中正確的是
.(請將所有正確命題的序號都填上)
查看答案和解析>>
科目:高中數學
來源:2011-2012學年廣東省外語外貿大學附設外語學校高三(上)數學寒假作業(yè)3(文科)(解析版)
題型:選擇題
關于平面向量

,

,

.有下列三個命題:
①若

•

=

•

,則

=

.
②若

=(1,k),

=(-2,6),

∥

,則k=-3.
③非零向量

和

滿足|

|=|

|=|

-

|,則

與

+

的夾角為60°.
其中真命題的個數有( )
A.0
B.1
C.2
D.3
查看答案和解析>>
科目:高中數學
來源:2011-2012學年山東省濰坊市現代中學高三(上)12月月考數學試卷(文科)(解析版)
題型:填空題
關于平面向量

,

,

,有下列三個命題:
①若

•

=

•

,則

=

、
②若

=(1,k),

=(-2,6),

∥

,則k=-3.
③非零向量

和

滿足|

|=|

|=|

-

|,則

與

+

的夾角為60°.
其中真命題的序號為
.(寫出所有真命題的序號)
查看答案和解析>>
科目:高中數學
來源:2008年陜西省高考數學試卷(文科)(解析版)
題型:解答題
關于平面向量

,

,

,有下列三個命題:
①若

•

=

•

,則

=

、
②若

=(1,k),

=(-2,6),

∥

,則k=-3.
③非零向量

和

滿足|

|=|

|=|

-

|,則

與

+

的夾角為60°.
其中真命題的序號為
.(寫出所有真命題的序號)
查看答案和解析>>
科目:高中數學
來源:2008年陜西省高考數學試卷(理科)(解析版)
題型:解答題
關于平面向量

,

,

,有下列三個命題:
①若

•

=

•

,則

=

、
②若

=(1,k),

=(-2,6),

∥

,則k=-3.
③非零向量

和

滿足|

|=|

|=|

-

|,則

與

+

的夾角為60°.
其中真命題的序號為
.(寫出所有真命題的序號)
查看答案和解析>>