题目内容
设平面向量
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A.(
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B.存在非零向量a,b同时满足
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C.(
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D.|
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【答案】分析:根据定义不难得出B是错误的,
⊙
=x1y2-y1x2=0,说明向
、
是互相平行的向量,若
•
=0,说明它们是垂直的向量.因为不存在两个非零向量,它们既平行又垂直,故B选项是错误的,而对于其它选项,可以分别证明它们是真命题.
解答:解:对于A,由定义得,
⊙
=x1y2-y1x2,
⊙
=x2y1-y2x1,所以(
⊙
)+(
⊙
)=0成立,A正确.
对于B,因为两个向量
、
平行的充要条件是x1y2-y1x2=0,若非零向量a,b同时满足
⊙
=0且
•
=0,说明两个向量既平行又垂直,故B选项是错误的.
设对于C,设
,则(
+
)⊙
=(x1+x2,y1+y2)⊙
=n(x1+x2)-m(y1+y2)=(nx1-y1m)+(nx2-my2)=
⊙
+
⊙
,故C选项是正确的.
对于D,|
⊙
|2=(x1y2-y1x2 )2=x12y22-2x1x2y1y2+y12x22
|
|2|
|2-|
•
|2=(x12+y12)(x22+y22)-(x1x2+y1y2)2=x12y22-2x1x2y1y2+y12y22,因此D选项是正确的.
故选B
点评:本题考查了在新定义下向量数量积的应用,属于基础题.牢记面向量的平行、垂直的充要条件,准确运用它们的坐标运算,是解决本题的关键.
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解答:解:对于A,由定义得,
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对于B,因为两个向量
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设对于C,设
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对于D,|
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|
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故选B
点评:本题考查了在新定义下向量数量积的应用,属于基础题.牢记面向量的平行、垂直的充要条件,准确运用它们的坐标运算,是解决本题的关键.
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