人教A版选修2-1第三章3.1.1空间向量及其加减运算达标过关训练

1、3.1.1空间向量及其加减运算一、选择题1对于空间中任意四点A，B，C，D，都有eq o(DA,sup6()eq o(CD,sup6()eq o(CB,sup6()()Aeq o(DB,sup6()Beq o(AC,sup6()Ceq o(AB,sup6() Deq o(BA,sup6()解析：eq o(DA,sup6()eq o(CD,sup6()eq o(CB,sup6()eq o(DA,sup6()eq o(CD,sup6()eq o(BC,sup6()eq o(BD,sup6()eq o(DA,sup6()eq o(BA,sup6().答案：D2在正方体ABCDA1B1C1D1中，下列

2、选项中化简后为零向量的是()Aeq o(AB,sup6()eq o(AD,sup6()eq o(AA1,sup6() Beq o(AB,sup6()eq o(AC,sup6()eq o(BB1,sup6()Ceq o(AB,sup6()eq o(A1D1,sup6()eq o(C1A1,sup6() Deq o(AC,sup6()eq o(CB1,sup6()eq o(AB,sup6()解析：eq o(AB,sup6()eq o(A1D1,sup6()eq o(C1A1,sup6()eq o(A1B1,sup6()eq o(A1D1,sup6()eq o(C1A1,sup6()eq o(A1C

3、1,sup6()eq o(A1C1,sup6()0.答案：C3如图所示，在正方体ABCDA1B1C1D1中，下列各式中运算结果为向量eq o(AC1,sup6()的是()(eq o(AB,sup6()eq o(BC,sup6()eq o(CC1,sup6()；(eq o(AA1,sup6()eq o(A1D1,sup6()eq o(D1C1,sup6()；(eq o(AB,sup6()eq o(BB1,sup6()eq o(B1C1,sup6()；(eq o(AA1,sup6()eq o(A1B1,sup6()eq o(B1C1,sup6().A BC D解析：(eq o(AB,sup6()e

4、q o(BC,sup6()eq o(CC1,sup6()eq o(AC,sup6()eq o(CC1,sup6()eq o(AC1,sup6().(eq o(AA1,sup6()eq o(A1D1,sup6()eq o(D1C1,sup6()eq o(AD1,sup6()eq o(D1C1,sup6()eq o(AC1,sup6().(eq o(AB,sup6()eq o(BB1,sup6()eq o(B1C1,sup6()eq o(AB1,sup6()eq o(B1C1,sup6()eq o(AC1,sup6().(eq o(AA1,sup6()eq o(A1B1,sup6()eq o(B1

5、C1,sup6()eq o(AB1,sup6()eq o(B1C1,sup6()eq o(AC1,sup6().答案：D4在下列条件中，使M与A、B、C一定共面的是()Aeq o(OM,sup6()3eq o(OA,sup6()2eq o(OB,sup6()eq o(OC,sup6()Beq o(OM,sup6()eq o(OA,sup6()eq o(OB,sup6()eq o(OC,sup6()0Ceq o(MA,sup6()eq o(MB,sup6()eq o(MC,sup6()0Deq o(OM,sup6()eq f(1,4)eq o(OB,sup6()eq o(OA,sup6()eq

6、f(1,2)eq o(OC,sup6()解析：在选项C中，eq o(MA,sup6()eq o(MB,sup6()eq o(MC,sup6()，故M与A，B，C四点共面答案：C5下列命题中正确的个数是()若a与b共线，b与c共线，则a与c共线；向量a，b，c共面即它们所在的直线共面；若ab，则存在唯一的实数，使ab.A1 B2C3 D0解析：中，b0时，则a与c不一定共线；中，由共面向量的定义是平行于同一平面的向量，表示这些向量的有向线段所在的直线不一定共面；中，当b0，a0时，不存在，故均错答案：D二、填空题6把所有单位向量的起点移到同一点，则这些向量的终点组成的图形是_解析：单位向量的长度

7、都相等，起点移到同一点，则终点在以单位长度为半径的球面上答案：球面7已知空间向量eq o(AB,sup6()，eq o(BC,sup6()，eq o(CD,sup6()，eq o(DA,sup6()，则下列结论正确的是_eq o(AB,sup6()eq o(BC,sup6()eq o(CD,sup6()；eq o(AB,sup6()eq o(DC,sup6()eq o(BC,sup6()eq o(AD,sup6()；eq o(AB,sup6()eq o(BC,sup6()eq o(DC,sup6()eq o(AD,sup6()；eq o(BC,sup6()eq o(BD,sup6()eq o(

8、DC,sup6().解析：eq o(BC,sup6()eq o(CD,sup6()eq o(BD,sup6()eq o(AB,sup6()，故不正确；eq o(AB,sup6()eq o(DC,sup6()eq o(BC,sup6()eq o(AB,sup6()eq o(BC,sup6()eq o(CD,sup6()eq o(AD,sup6()，故正确；不正确答案：8给出下列命题：方向相反的两个向量是相反向量；若A，B，C，D是不共线的四点，则eq o(AB,sup6()eq o(DC,sup6()是四边形ABCD为平行四边形的充要条件；|a|b|是向量ab的必要不充分条件；eq o(AB,s

9、up6()eq o(CD,sup6()的充要条件是A与C重合，B与D重合其中正确命题的序号为_解析：模相等，方向相反的两个向量是相反向量，故不正确，正确；模相等，方向相同的两个向量是相等向量，故不正确答案：三、解答题9设有空间四边形ABCD，对角线AC和BD的中点分别是E和F，求证：eq o(AB,sup6()eq o(CB,sup6()eq o(AD,sup6()eq o(CD,sup6()4eq o(EF,sup6().证明：如图，取CD中点M.则eq o(EF,sup6()eq o(EM,sup6()eq o(MF,sup6()eq f(1,2)eq o(AD,sup6()eq f(1,

10、2)eq o(CB,sup6()，eq o(AD,sup6()eq o(CB,sup6()2eq o(EF,sup6()，同理，可证eq o(AB,sup6()eq o(CD,sup6()2eq o(EF,sup6()，由得，eq o(AB,sup6()eq o(CB,sup6()eq o(AD,sup6()eq o(CD,sup6()4eq o(EF,sup6().10. 如图所示，在空间四边形OABC中，G，H分别是ABC，OBC的重心，设eq o(OA,sup6()a，eq o(OB,sup6()b，eq o(OC,sup6()c.试用向量a，b，c表示向量eq o(OG,sup6()和

11、eq o(GH,sup6().解：G为ABC重心，eq o(AG,sup6()eq f(2,3)eq o(AD,sup6()，eq o(AD,sup6()eq o(OD,sup6()eq o(OA,sup6().D为BC中点，eq o(OD,sup6()eq f(1,2)(eq o(OB,sup6()eq o(OC,sup6()，eq o(OG,sup6()eq o(OA,sup6()eq o(AG,sup6()eq o(OA,sup6()eq f(2,3)eq o(AD,sup6()eq o(OA,sup6()eq f(2,3)(eq o(OD,sup6()eq o(OA,sup6()eq o(OA,sup6()eq f(2,3)eq f(1,2)(eq o(OB,sup6()eq o(OC,sup6()eq f(2,3)eq o(OA,sup6()eq f(1,3)(eq o(OA,sup6()eq o(OB,sup6()eq o(OC,sup6()eq f(1,3)(abc)又eq o(OH,sup6()eq f(2,3)eq o(OD,sup6()eq f(2,3)eq