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www.ok6168.com:2019-2020年高中数学2.1向量的线性运算2.1.3向量的减法自我小测新人教B版必修

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2019-2020 年高中数学 2.1 向量的线性运算 2.1.3 向量的减法自我小测新
人教 B 版必修
1.如图所示,D,E,F 分别是△ABC 的边 AB,BC,CA 的中点,则-等于( )

A.

B.

C.

D.

2.在△ABC 中,||=||=|-|,则△ABC 是( )

A.直角三角形 B.钝角三角形 C.等边三角形 D.以上都不正确

3.已知平行四边形 ABCD,O 是平行四边形 ABCD 所在平面外任意一点,=a,=b,=c,则

向量等于( )

A.a+b+c B.a-b+c C.a+b-c D.a-b-c

4.在边长为 1 的正方形 ABCD 中,若=a,=b,=c,则|a-b+c|等于( )

A.0 B.1 C.2 D.

5.下列说法中错误的有( )

A.若+=,则-=

B.若+=,则+=

C.若+=,则-=

D.若+=,则+=

6.已知|a|=7,|b|=2,且 a∥b,则|a-b|=__________.

7.长度相等的三个非零向量,,满足++=0,则由 A,B,C 三点构成的△ABC 是________

三角形.

8.如图,已知向量 a,b,c,d,求作向量 a-b,c-d.

9.已知平面上不共线三点 A,B,C,O 是△ABC 内一点,若++=0,求证:O 是△ABC 的重 心.

参考答案 1.解析:由图可知=,=,则-=-=.又由三角形中位线定理,知=,故选 D.
答案:D 2.解析:因为-=,
所以||=||=||. 所以△ABC 是等边三角形. 答案:C 3.解析:如图,有=+=+=+-=a+c-b.
答案:B 4.解析:因为 c=a+b,所以|a-b+c|=|a+a|=|a|+|a|=2.
答案:C 5.答案:D 6.答案:5 或 9 7.解析:如图,作,的和向量.
因为++=0, 所以+=-. 所以=-. 又因为||=||=||=||, 所以△AOD 是等边三角形,四边形 AOBD 是菱形. 所以∠OAB=∠OAD=30°. 同理:∠OAC=∠OCA=∠OCB=∠OBC=∠OBA=30°. 所以∠BAC=∠ABC=∠ACB=60°, 即△ABC 为等边三角形. 答案:等边

8.解:如图,在平面内任取一点 O,作=a,=b,=c,=d,
则=a-b,=c-d. 9.证明:如图,因为++=0,所以=-(+),即+是的相反向量.
以 OB,OC 为邻边作?OBPC,则=+, 所以=-, 所以 A,O,P 共线. 又设 OP 与 BC 交 BC 于点 D, 则 D 是 BC 的中点, 所以 AD 是 BC 边上中线, 且 AD 过 O 点; 同理可证 BE,CF 分别是边 AC,AB 边上中线, 且都过 O 点. 所以 O 是△ABC 的重心.


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