新教材北师大版高中数学必修二 3.1向量的数乘运算 课时练（课后作业设计）

1、精品文档 精心整理精品文档 可编辑的精品文档向量的数乘运算1若abc，化简3(a2b)2(3bc)2(ab)的结果为()AaB4bCcDab2在ABC中，eq o(AB,sup8()c，eq o(AC,sup8()b，若点D满足eq o(BD,sup8()2eq o(DC,sup8()，则eq o(AD,sup8()()Aeq f(2,3)beq f(1,3)cBeq f(5,3)ceq f(2,3)bCeq f(2,3)beq f(1,3)cDeq f(1,3)beq f(2,3)c3在ABC中，M是BC的中点，则eq o(AB,sup8()eq o(AC,sup8()等于()Aeq f(1

2、,2)eq o(AM,sup8()Beq o(AM,sup8()C2eq o(AM,sup8()Deq o(MA,sup8()4已知ae12e2，b3e12e2，则3ab()A4e2B4e1C3e16e2D8e25若点O为平行四边形ABCD的中心，eq o(AB,sup8()2e1，eq o(BC,sup8()3e2，则eq f(3,2)e2e1()Aeq o(BO,sup8()Beq o(AO,sup8()Ceq o(CO,sup8()Deq o(DO,sup8()64(ab)3(ab)b等于_7在平行四边形ABCD中，eq o(AB,sup8()a，eq o(AD,sup8()b，eq o

3、(AN,sup8()3eq o(NC,sup8()，M为BC的中点，则eq o(MN,sup8()_.(用a，b表示)8已知点O为ABC外接圆的圆心，且eq o(OA,sup8()eq o(OB,sup8()eq o(CO,sup8()0，则ABC的内角A等于_9如图，四边形ABCD是一个梯形，eq o(AB,sup8()eq o(DC,sup8()且|eq o(AB,sup8()|2|eq o(DC,sup8()|，M，N分别是DC，AB的中点，已知eq o(AB,sup8()e1，eq o(AD,sup8()e2，试用e1，e2表示下列向量(1)eq o(AC,sup8()；(2)eq o

4、(MN,sup8().10已知E，F分别为四边形ABCD的对角线AC，BD的中点，设eq o(BC,sup8()a，eq o(DA,sup8()b，试用a，b表示eq o(EF,sup8().11设D，E，F分别是ABC三边BC，CA，AB的中点，则eq o(DA,sup8()2eq o(EB,sup8()3eq o(FC,sup8()()Aeq f(1,2)eq o(AD,sup8()Beq f(3,2)eq o(AD,sup8()Ceq f(1,2)eq o(AC,sup8()Deq f(3,2)eq o(AC,sup8()12在平行四边形ABCD中，AC与BD交于点O，E是线段OD的中点

5、，AE的延长线与CD交于点F，若eq o(AC,sup8()a，eq o(BD,sup8()b，则eq o(AF,sup8()等于()Aeq f(1,4)aeq f(1,2)bBeq f(1,3)aeq f(2,3)bCeq f(1,2)aeq f(1,4)bDeq f(2,3)aeq f(1,3)b13设a、b都是非零向量，下列四个条件中，使eq f(a,|a|)eq f(b,|b|)成立的充分条件是()A|a|b|且abBabCabDa2b14在ABC中，已知D是AB边上一点，若eq o(AD,sup8()2eq o(DB,sup8()，eq o(CD,sup8()eq f(1,3)eq

6、o(CA,sup8()eq o(CB,sup8()，则的值为_15如图，AB是O的直径，点C，D是半圆弧AB上的两个三等分点，试用eq o(AB,sup8()，eq o(AC,sup8()表示eq o(AD,sup8().答案1若abc，化简3(a2b)2(3bc)2(ab)的结果为()AaB4bCcDabA3(a2b)2(3bc)2(ab)(32)a(662)b2ca2(bc)a2aa.2在ABC中，eq o(AB,sup8()c，eq o(AC,sup8()b，若点D满足eq o(BD,sup8()2eq o(DC,sup8()，则eq o(AD,sup8()()Aeq f(2,3)beq

7、 f(1,3)cBeq f(5,3)ceq f(2,3)bCeq f(2,3)beq f(1,3)cDeq f(1,3)beq f(2,3)cAeq o(BD,sup8()2eq o(DC,sup8()，eq o(AD,sup8()eq o(AB,sup8()2(eq o(AC,sup8()eq o(AD,sup8()，3eq o(AD,sup8()2eq o(AC,sup8()eq o(AB,sup8()eq o(AD,sup8()eq f(2,3)eq o(AC,sup8()eq f(1,3)eq o(AB,sup8()eq f(2,3)beq f(1,3)c.3在ABC中，M是BC的中点

8、，则eq o(AB,sup8()eq o(AC,sup8()等于()Aeq f(1,2)eq o(AM,sup8()Beq o(AM,sup8()C2eq o(AM,sup8()Deq o(MA,sup8()C作平行四边形ABEC，则M是对角线的交点，故M是AE的中点，由题意知，eq o(AB,sup8()eq o(AC,sup8()eq o(AE,sup8()2eq o(AM,sup8()，故选C4已知ae12e2，b3e12e2，则3ab()A4e2B4e1C3e16e2D8e2D3ab3(e12e2)(3e12e2)3e16e23e12e28e2.5若点O为平行四边形ABCD的中心，eq

9、 o(AB,sup8()2e1，eq o(BC,sup8()3e2，则eq f(3,2)e2e1()Aeq o(BO,sup8()Beq o(AO,sup8()Ceq o(CO,sup8()Deq o(DO,sup8()A因为eq o(BD,sup8()eq o(AD,sup8()eq o(AB,sup8()eq o(BC,sup8()eq o(AB,sup8()3e22e1，所以eq o(BO,sup8()eq f(1,2)eq o(BD,sup8()eq f(3,2)e2e1.64(ab)3(ab)b等于_a8b原式4a4b3a3bba8b.7在平行四边形ABCD中，eq o(AB,sup

10、8()a，eq o(AD,sup8()b，eq o(AN,sup8()3eq o(NC,sup8()，M为BC的中点，则eq o(MN,sup8()_.(用a，b表示)eq f(1,4)beq f(1,4)aeq o(MN,sup8()eq o(MB,sup8()eq o(BA,sup8()eq o(AN,sup8()eq f(1,2)baeq f(3,4)eq o(AC,sup8()eq f(1,2)baeq f(3,4)(ab)eq f(1,4)beq f(1,4)a.8已知点O为ABC外接圆的圆心，且eq o(OA,sup8()eq o(OB,sup8()eq o(CO,sup8()0，

11、则ABC的内角A等于_30由eq o(OA,sup8()eq o(OB,sup8()eq o(CO,sup8()0得，eq o(OA,sup8()eq o(OB,sup8()eq o(OC,sup8()，结合向量加法的几何意义知四边形OACB为平行四边形，又OAOB，则四边形OACB为菱形，所以OAC是正三角形，所以CAO60所以CABeq f(1,2)CAO309如图，四边形ABCD是一个梯形，eq o(AB,sup8()eq o(DC,sup8()且|eq o(AB,sup8()|2|eq o(DC,sup8()|，M，N分别是DC，AB的中点，已知eq o(AB,sup8()e1，eq

12、o(AD,sup8()e2，试用e1，e2表示下列向量(1)eq o(AC,sup8()；(2)eq o(MN,sup8().解因为eq o(AB,sup8()eq o(DC,sup8()，|eq o(AB,sup8()|2|eq o(DC,sup8()|，所以 eq o(AB,sup8()2eq o(DC,sup8()，eq o(DC,sup8()eq f(1,2)eq o(AB,sup8().(1)eq o(AC,sup8()eq o(AD,sup8()eq o(DC,sup8()e2eq f(1,2)e1.(2)eq o(MN,sup8()eq o(MD,sup8()eq o(DA,su

13、p8()eq o(AN,sup8()eq f(1,2)eq o(DC,sup8()eq o(AD,sup8()eq f(1,2)eq o(AB,sup8()eq f(1,4)e1e2eq f(1,2)e1eq f(1,4)e1e2.10已知E，F分别为四边形ABCD的对角线AC，BD的中点，设eq o(BC,sup8()a，eq o(DA,sup8()b，试用a，b表示eq o(EF,sup8().解如图所示，取AB的中点P，连接EP，FP.在ABC中，EP是中位线，所以eq o(PE,sup8()eq f(1,2)eq o(BC,sup8()eq f(1,2)a.在ABD中，FP是中位线，所

14、以eq o(PF,sup8()eq f(1,2)eq o(AD,sup8()eq f(1,2)eq o(DA,sup8()eq f(1,2)b.在EFP中，eq o(EF,sup8()eq o(EP,sup8()eq o(PF,sup8()eq o(PE,sup8()eq o(PF,sup8()eq f(1,2)aeq f(1,2)beq f(1,2)(ab)11设D，E，F分别是ABC三边BC，CA，AB的中点，则eq o(DA,sup8()2eq o(EB,sup8()3eq o(FC,sup8()()Aeq f(1,2)eq o(AD,sup8()Beq f(3,2)eq o(AD,su

15、p8()Ceq f(1,2)eq o(AC,sup8()Deq f(3,2)eq o(AC,sup8()DD，E，F分别是ABC三边BC，CA，AB的中点，eq o(DA,sup8()2eq o(EB,sup8()3eq o(FC,sup8()eq f(1,2)eq blc(rc)(avs4alco1(o(BA,sup8()o(CA,sup8()2eq blc(rc)(avs4alco1(f(o(AB,sup8()o(CB,sup8(),2)3eq blc(rc)(avs4alco1(f(o(AC,sup8()o(BC,sup8(),2)eq f(1,2)eq o(BA,sup8()eq f(

16、1,2)eq o(CA,sup8()eq o(AB,sup8()eq o(CB,sup8()eq f(3,2)eq o(AC,sup8()eq f(3,2)eq o(BC,sup8()eq f(1,2)eq o(AB,sup8()eq o(AC,sup8()eq f(1,2)eq o(BC,sup8()eq f(1,2)eq o(AC,sup8()eq o(AC,sup8()eq f(3,2)eq o(AC,sup8()12在平行四边形ABCD中，AC与BD交于点O，E是线段OD的中点，AE的延长线与CD交于点F，若eq o(AC,sup8()a，eq o(BD,sup8()b，则eq o(A

17、F,sup8()等于()Aeq f(1,4)aeq f(1,2)bBeq f(1,3)aeq f(2,3)bCeq f(1,2)aeq f(1,4)bDeq f(2,3)aeq f(1,3)bDDEFBEA，eq f(DF,AB)eq f(DE,EB)eq f(1,3)，DFeq f(1,3)AB，eq o(AF,sup8()eq o(AD,sup8()eq o(DF,sup8()eq o(AD,sup8()eq f(1,3)eq o(AB,sup8().eq o(AC,sup8()eq o(AB,sup8()eq o(AD,sup8()a，eq o(BD,sup8()eq o(AD,sup8

18、()eq o(AB,sup8()b，联立得：eq o(AB,sup8()eq f(1,2)(ab)，eq o(AD,sup8()eq f(1,2)(ab)，eq o(AF,sup8()eq f(1,2)(ab)eq f(1,6)(ab)eq f(2,3)aeq f(1,3)b.13设a、b都是非零向量，下列四个条件中，使eq f(a,|a|)eq f(b,|b|)成立的充分条件是()A|a|b|且abBabCabDa2bD由a2b，得eq f(a,|a|)eq f(2b,|2b|)eq f(2b,2|b|)eq f(b,|b|)，故选D14在ABC中，已知D是AB边上一点，若eq o(AD,sup8()2eq o(DB,sup8()，eq o(CD,sup8()eq f(1,3)eq o(CA,sup8()eq o(CB,sup8()，则的值为_eq f(2,3)eq o(CD,sup8()eq o(CA,sup8()eq o(AD,sup8()eq o(CA,sup8()eq f(2,3)eq o(AB,sup8()eq o(CA,sup8()eq f(2,3)(eq o(CB,sup8()eq o(CA,sup8()eq f(1,3)eq o(CA,sup8()eq f(2,3)eq o(CB,s