平面向量及坐标表示

平面向量及坐标表示 授课主题 平面向量的基本定理及坐标表示 1、了解平面向量的基本定理及其意义，会用平面向量基本定理解决简单问题( 教学目的 2(掌握平面向量的正交分解及其坐标表示( 1(会用坐标表示平面向量的加法、减法与数乘运算( 教学重点 2(理解用坐标表示的平面向量共线的条件 教学内容 1(平面向量基本定理 定理:如果e，e是同一平面内的两个__________向量，那么对于这一平面内的任意向量a，12 __________一对实数λ，λ，使a,__________，其中，__________叫做表示这一平面内所有向12 量的一组基底，记为{e，e}( 12 2(平面向量的坐标表示 (1)在平面直角坐标系中，分别取与x轴、y轴方向相同的两个单位向量i，j作为基底，对于平面内的一个向量a，有且只有一对实数x，y使a,xi，yj，把有序数对__________叫做向量a的坐标，记作a,____，其中__________叫做a在x轴上的坐标，__________叫做a在y轴上的坐标，显然0,(0,0)，i,(1,0)，j,(0,1)( ??(2)设OA,xi，yj，则__________就是终点A的坐标，即若OA,(x，y)，则A点坐标为(x，y)，反之亦成立(O是坐标原点)( 3(平面向量的坐标运算 (1)加法、减法、数乘运算 a b λa 向量 a，b a,b 坐标 (x，y) (x，y) (x，x，y，y) (x,x，y,y) (λx，λy) 11221212121211(2)向量坐标的求法 ?已知A(x，y)，B(x，y)，则AB,__________，即一个向量的坐标等于__________( 1122 (3)平面向量共线的坐标表示 设a,(x，y)，b,(x，y)， 1122 其中b?0，则a与b共线?a,__________?__________. a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 1(若a,(3,2)，b,(0，,1)，则2b,a的坐标是( )( A((3，,4) B((,3,4) C((3,4) D((,3，,4) 22(已知向量a,(1，,m)，b,(m，m)，则向量a，b所在的直线可能为( )( A(x轴 B(第一、三象限的角平分线 C(y轴 D(第二、四象限的角平分线 3(已知a,(4,5)，b,(8，y)且a?b，则y等于( )( 32A(5 B(10 C( D(15 5 4(e，e是平面内一组基底，那么( )( 12 A(若实数λ，λ使λe，λe,0，则λ,λ,0 12112212 B(空间内任一向量a可以表示为a,λe，λe(λ，λ为实数) 112212C(对实数λ，λ，λe，λe不一定在该平面内 121122 D(对平面内任一向量a，使a,λe，λe的实数λ，λ有无数对 112212 一、平面向量基本定理 ????【例1】已知梯形ABCD，如图所示，2DC,AB，M，N分别为AD，BC的中点(设AD,e，AB1 ???,e，试用e，e表示DC，BC，MN. 212 变式练习 ??1、(2012大纲全国高考)?ABC中，AB边的高为CD，若CB,a，CA,b， a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, ?a?b,0，|a|,1，|b|,2，则AD,( )( 1122A(a,b B(a,b 3333 3344C(a,b D(a,b 5555 12?2、 (1)(2013?江苏)设D，E分别是?ABC的边AB，BC上的点，AD,AB，BE,BC.若DE,23??λAB，λAC(λ，λ为实数)，则λ，λ的值为________( 121212 ???????(2)?ABC的外接圆的圆心为O，半径为2，OA，AB，AC,0且|OA|,|AB|，则向量CA在CB 上的投影为 ( ) A.3 B(3 C(,3 D(,3 1答案 (1) (2)A 2 1212????????解析 (1)如图～DE,DB，BE,AB，BC,AB，(AC,AB) 2323 12121??,,AB，AC～则λ,,～λ,～λ，λ,. 121263632 ???(2)由OA，AB，AC,0～ ???得AB，AC,AO. 又O为?ABC外接圆的圆心～OB,OC～ ?四边形ABOC为菱形～AO?BC. ??由|OA|,|AB|,2～ 知?AOC为等边三角形( π???故CA在CB上的投影为|CA|cos?ACB,2cos ,3. 6 二、平面向量的坐标运算 ???【例2】已知A(,2,4)，B(3，,1)，C(,3，,4)(设AB,a，BC,b，CA,c. (1)求3a，b,3c; (2)求满足a,mb，nc的实数m，n. 3 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 变式练习 ?????在?ABC中，点P在BC上，且BP,2PC，点Q是AC的中点，若PA,(4,3)，PQ,(1,5)，则BC等于( )( A((,6,21) B((,2,7) C((6，,21) D((2，,7) 三、平面向量共线的坐标表示 【例3,1】已知向量a,(1,2)，b,(1,0)，c,(3,4)(若λ为实数，(a，λb)?c，则λ,( )( 11A( B( C(1 D(2 42 【例3,2】已知a,(1,0)，b,(2,1)， (1)当k为何值时，ka,b与a，2b共线; ??(2)若AB,2a，3b，BC,a，mb且A，B，C三点共线，求m的值( a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 方法提炼 向量共线的坐标表示既可以判定两向量平行，也可以由平行求参数(当两向量的坐标均非零时，也可以利用坐标对应成比例来求解( 提醒:若a,(x～y)～b,(x～y)～则a?b的充要条件不能表示1122 xy11成,～因为x～y有可能等于0～所以应表示为xy,xy,0.同时～221221xy22 a?b的充要条件也不能错记为:xx,yy,0～xy,xy,0等( 12121122 变式练习 ,3,,11,1、设a,,sin x，,，b,,，cos x,，且a?b，则锐角x等于( )( 432,,,, πππ5πA( B( C( D( 64312 2、已知直角坐标平面内的两个向量a,(1,3)，b,(m,2m,3)，使得平面内的任意一个向量c都可以唯一的表示成c,λa，μb，则m的取值范围是__________( 四、平面向量基本定理的应用 例 如图，已知?ABC的面积为14，D、E分别为边AB、BC上的点，且AD?DB,BE?EC, ??????2?1，AE与CD交于P.设存在λ和μ使AP,λAE，PD,μCD，AB,a，BC,b. (1) 求λ及μ; ?(2) 用a、b表示BP; (3) 求?PAC的面积( 21????解:(1) 由于AB,a～BC,b～则AE,a，b～DC,a，b. 33 5 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 21,,,,????,,,,AP,λAE,λa，b～DP,μDC,μa，b～ ,,,,33 2212,,?????,,AP,AD，DP,AB，DP～即a，μ(a，b),λa，b. ,,3333 21λ,，μ～,,3364 解得λ,～μ,. ,772μ,λ～,,3 6214,,???,,(2) BP,BA，AP,,a，a，b,,a，b. ,,7377 (3) 设?ABC、?PAB、?PBC的高分别为h、h、h～ 12 44??h?h,|PD|?|CD|,μ,～S,S,8. ??1PABABC77 11??h?h,|PE|?|AE|,1,λ,～S,S,2～ ??2PBCABC77 ? S,4. ?PAC ??如图所示，在?ABC中，H为BC上异于B、C的任一点，M为AH的中点，若AM,λAB?，μAC，则λ，μ,________( 1答案: 2 ????解析:由B、H、C三点共线～可令AH,xAB，(1,x)AC～又M是AH的中点～所以AM, 111???AH,xAB，(1,x)AC. 222 111???又AM,λAB，μAC～所以λ，μ,x，(1,x),. 222 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, ??????1、已知?ABC和点M满足MA，MB，MC,0.若存在实数m使得AB，AC,mAM成立，则m的 值为 ( ) A(2 B(3 C(4 D(5 ?????2、如图，平面内有三个向量OA，OB，OC，其中OA与OB的夹角为120?， ????????OA与OC的夹角为30?，且|OA|,|OB|,1，|OC|,23，若OC,λOA，μOB (λ，μ?R)，则λ，μ的值为________( 答案 (1)B (2)6 ???解析 (1)?MA，MB，MC,0～?点M是?ABC的重心( ????AB，AC,3AM～?m,3. ??????(2)方法一 如图～OC,OB，OA～|OB|,2～|OA|,|BC|,4～ 11111 ????OC,4OA，2OB. ?λ，μ,6. 3、设平面向量a,(,1,0)，b,(0,2)，则2a,3b,( ) A((6,3) B((,2，,6) C((2,1) D((7,2) 解析:2a,3b,(,2,0),(0,6),(,2，,6)( 答案:B 24、已知平面向量a,(x,1)，b,(,x，x)，则向量a，b( )( A(平行于x轴 B(平行于第一、三象限的角平分线 C(平行于y轴 D(平行于第二、四象限的角平分线 22解析 由题意得a，b,(x,x,1，x),(0,1，x)，易知a，b平行于y轴( 7 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 答案 C 5、已知平面向量a,(1,2)，b,(,2，m)，且a?b，则2a，3b,( )( A((,2，,4) B((,3，,6) C((,4，,8) D((,5，,10) 解析 由,(1,2)，,(,2，)，且?，得1×,2×(,2)?,,4，从而,(,2，,abmabmmb4)，那么2a，3b,2×(1,2)，3×(,2，,4),(,4，,8)( 答案 C ??6、 设点A(2,0)，B(4,2)，若点P在直线AB上，且|AB|,2|AP|，则点P的坐标为( ) A((3,1) B((1，,1) C((3,1)或(1，,1) D(无数多个 ????????解析 设P(x，y)，则由|AB|,2|AP|，得AB,2AP或AB,,2AP，AB,(2,2)，AP,(x,2，y)，即(2,2),2(x,2，y)，x,3，y,1，P(3,1)，或(2,2),,2(x,2，y)，x,1，y,,1， P(1，,1)( 答案 C 117、若三点(2,2)，(0)，(0，)(?0)共线，则，的值为________( ABa,Cbabab ?? 解析 AB,(a,2，,2)，AC,(,2，b,2)，依题意，有(a,2)(b,2),4,0， 111即ab,2a,2b,0，所以，,. ab2 8、设向量a，b满足|a|,25，b,(2,1)，且a与b的方向相反，则a的坐标为________( 解析 设a,λb(λ,0)，则|a|,|λ||b|， |a|?|λ|,， |b| 又|b|,5，|a|,25. ?|λ|,2，?λ,,2. ?a,λb,,2(2,1),(,4，,2)( 答案 (,4，,2) 9、设e，e是平面内一组基向量，且a,e，2e，b,,e，e，则向量e，e可以表示为另一12121212组基向量a，b的线性组合，即e，e,________a，________b. 12 解析 由题意，设e，e,ma，nb. 12 又因为a,e，2e，b,,e，e，所以e，e,m(e，2e)，n(,e，e),(m,n)e，(2m，n)e. 121212121212 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 2,m,，,m,n,1，3,由平面向量基本定理，得,所以 , 2m，n,1，1, n,,.,,3 10、在平面直角坐标系xOy中，四边形ABCD的边AB?DC，AD?BC.已知点A(,2,0)，B(6,8)，C(8,6)，则D点的坐标为________( 解析 由条件中的四边形ABCD的对边分别平行，可以判断该四边形ABCD是平行四边形(设D(x， ?? y)，则有AB,DC，即(6,8),(,2,0),(8,6),(x，y)，解得(x，y),(0，,2)( 答案 (0，,2) 11(已知A(1,1)、B(3，,1)、C(a，b)( (1)若、、三点共线，求、的关系式; ABCab ,,,,,,,, AC(2)若,2，求点C的坐标( AB ,,,,,,,, AC解析:(1)由已知得,(2，,2)，,(a,1，b,1)， AB ,,,,,,,, AC?、、三点共线，??. ABCAB ?2(b,1)，2(a,1),0，即a，b,2. ,,,,,,,, AC(2)?,2， AB ?(a,1，b,1),2(2，,2)， ,a,1,4，,a,5，?,解得, b,1,,4，b,,3.,, ?点C的坐标为(5，,3)( ??? 12(已知O(0,0)，A(1,2)，B(4,5)及OP,OA，tAB，求 (1)t为何值时，P在x轴上,P在y轴上,P在第二象限, (2)四边形OABP能否成为平行四边形,若能，求出相应的t值;若不能，请说明理由( ???2解析 (1)OP,OA，tAB,(1，3t,2，3t)(若P在x轴上，则2，3t,0，?t,,;若P在y轴3 1，3t,0，,1上，只需1，3t,0，?t,,;若P在第二象限，则, 32，3t,0., 9 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, 21?,,t,,. 33 ????,3,3t,1，(2)因为OA,(1,2)，PB,(3,3t,3,3t)(若OABP为平行四边形，则OA,PB，?, 3,3t,2, 无解(所以四边形OABP不能成为平行四边形 参考答案 基础梳理自测 知识梳理 1(不共线 有且只有 λe，λe 不共线的向量e，e 112212 2((1)(x，y) (x，y) x y ,,,, (2)向量OA的坐标 3((2)(x,x，y,y) 终点的坐标减去起点的坐标 2121 (3)λb xy,xy,0 1221 基础自测 1(D 解析:?2b,a,2×(0～,1),(3,2),(0～,2),(3,2),(,3～,4)～ 故2b,a,(,3～,4)( 222(A 解析:a，b,(1～,m)，(m～m),(m，1,0)(其横坐标恒大于零～纵坐标等于零～故向量a，b 所在的直线可能为x轴( 3(B 解析:?a?b～ ?4y,40,0～得y,10. 4(A 解析:对于A～?e～e不共线～故λ,λ,0正确, 1212 对于B～空间向量a应改为与e～e共面的向量才可以, 12 C中～λe，λe一定与e～e共面, 112212 D中～根据平面向量基本定理～λ～λ应是唯一一对( 12 考点探究突破 ,,,,,,,,,,,, DCDC【例1】解:?2,AB～?2,e～ 2,,,,1DC?,e. 22,,,,,,,,,,,,,,,, BCDC又?,BA，AD，～ ,,,,11BC?,,e，e，e,e,e. 2121222,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,111113MNBNMNBCMAABDAAB又由,，，～得,，，,,e，e，(e,e),e. 12122222224【例2】解:由已知得a,(5～,5)～ b,(,6～,3)～c,(1,8)( (1)3a，b,3c,3(5～,5)，(,6～,3),3(1,8),(15,6,3～,15,3,24),(6～,42)( (2)?mb，nc,(,6m，n～,3m，8n),(5～,5)～ ,,,6m，n,5～m,,1～,,,,?解得 ,,3m，8n,,5.,n,,1.,, 【例3,1】B 解析:?a,(1,2)～b,(1,0)～c,(3,4)～ ?a，λb,(1,2)，(λ～0),(1，λ～2)( 又?(a，λb)?c～ 1，λ21?,～解得λ,. 342 【例3,2】解:(1)ka,b,k(1,0),(2,1),(k,2～,1)～a，2b,(1,0)，2(2,1),(5,2)( a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, ?ka,b与a，2b共线～ ?2(k,2),(,1)×5,0～ 1即2k,4，5,0～得k,,. 2,,,,,,,,(2)?A～B～C三点共线～?存在实数λ～使,λ～ BCAB即2a，3b,λ(a，mb)( ,2,λ～,3,?解得m,. 23,mλ.,, 演练巩固提升 1(D 解析:?a?b,0～?a?b. 又?|a|,1～|b|,2～ ,,,, ?||,5～ AB,,,,1×225CD?||,,. 55 ,,,,452252,,?||,2,,. AD,,55 45,,,,,,,,,,,,54444?,,,(a,b),a,b. ADABAB55555,,,,,,,,,,,,,,,,,,,,,,,,,,,, PC2(A 解析:如图～,,,,(1,5),(4,3),(,3,2)～,，,(1,5)，(,3,2)PAQCAQPQPQQC ,(,2,7)～ ,,,,,,,, BCPC,3,(,6,21)( 113,,,,3(B 解析:?a,sin x～～b,～cos x～且a?b～ ,,,,432131?sin xcos x,×,0～ 243 11即sin 2x,,0.?sin 2x,1. 44 ππ又?x为锐角～?2x,～x,. 24 4({m|m?,3} 解析:要使c,λa，μb成立～ 则只需a与b不共线即可～ 2m,3m?只需满足?～ 13 即3m?2m,3～?m?,3. 11 a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou, a variety of connectors, supporting or providing temporary support. Lateral template installation, install a waterproof top bracing, clean the wall of debris, then install the template in place on the inside, the template is in place to ensure the template bottom line with the elastic edge of the template, template template control lines when the lines are not visible (parallel with the template invocation line 20 cm) check the template in place. Accessories must be inserted firmly, u-shaped card on the same seam should not be clamped in the same direction. D. support: templates in place immediately after the installation template supporting structure, initially adjust the verticality of the template, and then follow the advance reserved hole insert the bolt in the template. Wall bolts should be straight, interspersed with cable tension bolts, hard top. Pillar and bracing support surface should be flat pad solid, to have enough pressure area. Support should focus on for the steel cord. 3, and beam, and roof construction (1) main process: play axis, and elevation control line ? beam, and Board template support erection ? Latrun line installation beam end of pipe ? installation Liang Dimo, and brush off touch agent ? lashing beam steel ? beam steel hidden qualified ? installation beam side die and reinforcement ? Latrun line installation steel beams ? placed wood Fang ? tile multilayer Board ? template gap processing ? uniform brushing release Agent ? template pre check (2) construction points: a,. stay wall body template demolition Hou,