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1、2013年高中数学解题思路大全：空间向量与立体几何解答题精选(数学选修2-1)第三章 空间向量与立体几何解答题精选 1 已知四棱锥的底面为直角梯形，PABCD,ABDC/1,底面，且，ABCDPAADDC,，DAB,90,PA,2，是的中点 AB,1MPB(?)证明:面面; PCDPAD,(?)求与所成的角; ACPB(?)求面与面所成二面角的大小 AMCBMC证明:以为坐标原点长为单位长度，如图建立空间直角坐标系，则各点坐标为 AAD1 ABCDPM(0,0,0),(0,2,0),(1,1,0),(1,0,0),(0,0,1),(0,1,)2(?)证明:因 AP,(0,0,1),DC,(0,
2、1,0),故AP,DC,0,所以AP,DC.由题设知，且与是平面内的两条相交直线，由此得面ADDC,DC,APADPAD又在面上，故面?面 DCPCDPCDPADPAD(?)解:因 AC,(1,1,0),PB,(0,2,1),故|AC|,2,|PB|,5,AC,PB,2,所以AC,PB10cos,AC,PB,.5|AC|,|PB|MC(?)解:在上取一点，则存在使 NC,MC,Nxyz(,),R,11 NC,(1,x,1,y,z),MC,(1,0,),？x,1,y,1,z,.22,14要使,只需即解得 , ANMCANMCxz,00,.25412,可知当,时,N点坐标为(,1,),能使AN,M
3、C,0.555 1212此时,AN,(,1,),BN,(,1,),有BN,MC,05555由AN,MC,0,BN,MC,0得AN,MC,BN,MC.所以，ANB为 所求二面角的平面角 ,30304？ |,|,.ANBNANBN,555,ANBN 2,？,cos(,).ANBN 3|ANBN,2故所求的二面角为arccos().,3高考学习网,中国最大高考学习网站G | 我们负责传递知识 2 如图，在四棱锥中，底面是正方形，侧面是正三角形, VABCD,ABCDVAD平面底面 VAD,ABCD(?)证明:平面; VADAB,(?)求面与面所成的二面角的大小 VADDB证明:以为坐标原点，建立如图
4、所示的坐标图系 D(?)证明:不防设作， A(1,0,0)13则， ， V(,0,)B(1,1,0)2213 AB,(0,1,0),VA,(,0,)22由得，又，因而与平面内两条相交直线ABVA,VADAB,VA,0,ABAD,AB，都垂直 ?平面 VAVADADAB,13 (?)解:设为DV中点，则， E(,0,)E44333313 EA,(,0,),EB,(,1,),DV,(,0,).444422由 EB,DV,0,得EB,DV,又EA,DV.因此，是所求二面角的平面角， ，AEBEA,EB21 cos(EA,EB),7|EA|,|EB|21解得所求二面角的大小为 arccos.7PABC
5、D,ABCD3 如图，在四棱锥中，底面为矩形， VAB,3ABCDBC,1侧棱底面， PA,PA,2CD为的中点 EPDAC (?)求直线与所成角的余弦值; PBABNNE,PAC(?)在侧面内找一点，使面， PAB N并求出点到和的距离 ABAP解:(?)建立如图所示的空间直角坐标系， 则的坐标为、 ABCDPE,A(0,0,0)、 B(3,0,0)C(3,1,0)D(0,1,0)高考学习网,中国最大高考学习网站G | 我们负责传递知识 1、， E(0,1)P(0,0,2)2从而 AC,(3,1,0),PB,(3,0,2).设的夹角为，则 AC与PB,AC,PB337cos, 1427|AC
6、|,|PB|37?与所成角的余弦值为 ACPB14(?)由于点在侧面内，故可设点坐标为，则 NNPAB(,0,)xz1，由面可得， NE,PACNE,(,x,1,z)2,1,3z,1,0,(,x,1,z),(0,0,2),0,x,NE,AP,0,2 ? ,6即化简得,11,3x，,0.,NE,AC,0.,z,1,(,x,1,z),(3,1,0),0.,2,2,33NN即点的坐标为，从而点到和的距离分别为 (,0,1)ABAP1,66ABCDAECF4 如图所示的多面体是由底面为的长方体被截面所截面而得到的，其中 1ABBCCCBE,4,2,3,1 1(?)求的长; BFCAECF (?)求点到
7、平面的距离 1解:(I)建立如图所示的空间直角坐标系，则， D(0,0,0)B(2,4,0)ACEC(2,0,0),(0,4,0),(2,4,1),(0,4,3)设 Fz(0,0,)1AECF?为平行四边形， 1高考学习网,中国最大高考学习网站G | 我们负责传递知识 ？由AECF为平行四边形,1？由AF,EC得,(,2,0,z),(,2,0,2),1？z,2.？F(0,0,2).？EF,(,2,4,2).于是|BF|,26,即BF的长为26.(II)设为平面的法向量， AECFn11显然n不垂直于平面ADF,故可设n,(x,y,1)11,n,AE,0,0，x，4，y，1,0,1由得 ,xy,
8、2，0，2,0,n,AF,0,1,x,1,4y，1,0, 即？,1,2x，2,0,y,.,4,的夹角为，则 又CC,(0,0,3),设CC与n111CC,n343311cos,. 331|CC|,|n|113，1，116CAECF?到平面的距离为 1433433d,|CC|cos,3，,. 13311ABCDABCD,ADAAAB,1,25 如图，在长方体，中，点在棱上移EAD11111DEAD,动 (1)证明:; 11ACD (2)当为的中点时，求点到面的距离; EABE1,DECD, (3)等于何值时，二面角的大小为 AE14高考学习网,中国最大高考学习网站G | 我们负责传递知识 解:以
9、为坐标原点，直线分别为轴，建立空间直角坐标系，DADCDD,Dxyz,1设，则 AEx,ADExAC(1,0,1),(0,0,1),(1,0),(1,0,0),(0,2,0)11(1) 因为DA,DE,(1,0,1),(1,x,1),0,所以DA,DE.1111(2)因为为的中点，则，从而， DE,(1,1,1),AC,(,1,2,0)EABE(1,1,0)1,n,AC,0,，设平面的法向量为，则也即ACDAD,(,1,0,1)n,(a,b,c),11,n,AD,0,1,a,2b,a，2b,0,，得，从而，所以点到平面的距离为 ACDn,(2,1,2)E,1,a，c,0a,c,|DE,n|2，
10、1,211h,. 33|n|(3)设平面DEC的法向量，?n,(a,b,c)1CE,(1,x,2,0),DC,(0,2,1),DD,(0,0,1),11,n,DC,0,2b,c,0,1,由 令， bcax,？,1,2,2,abx，(,2),0.,n,CE,0,? n,(2,x,1,2).|n,DD|,2221cos,.依题意 2422|n|,|DD|(x,2)，51?(不合，舍去)， x,2，3x,2,312,AE,23DECD,?时，二面角的大小为 14ABCABC,BBCCCCCC,6 如图，在三棱柱中，侧面，为棱上异于的AB,E1111111,EAEB,ABBBBCBCC,，,2,2,1
11、,一点，已知，求: 1113EB (?)异面直线与的距离; AB1AEBA, (?)二面角的平面角的正切值 11高考学习网,中国最大高考学习网站G | 我们负责传递知识 解:(I)以为原点，、分别为轴建立空间直角坐标系 BBBAByz,1, 由于， ABBBBCBCC,，,2,2,1,113在三棱柱中有 ABCABC,1113133 , BAB(0,0,0),(0,0,2),(0,2,0)C(,0),C(,0)1122223 设 E(,a,0),由EA,EB,得EA,EB,0,即11233 0,(,a,2),(,2,a,0)22332 ,，a(a,2),a,2a，,44131331得(a,)(
12、a,),0,即a,或a,(舍去),故E(,0)222222 313333BE,EB,(,0),(,0),，,0,即BE,EB.11222244又BBCCABEB,侧面，故 因此是异面直线的公垂线， AB,ABBE,BE11131|BE|,，,1ABEB,则，故异面直线的距离为 1144,AEBA,(II)由已知有故二面角的平面角的大小为向EA,EB,BA,EB,111111量的夹角 BA与EA1131因BA,BA,(0,0,2),EA,(,2),1122EA,BA211,故cos, 3|EA|BA|112即tan,.2PABCD,ABCDABCD7 如图，在四棱锥中，底面为矩形，PD,底面，E
13、是AB上 1PFEC,PD,2,CD,2,AE,一点， 已知 2高考学习网,中国最大高考学习网站G | 我们负责传递知识 求(?)异面直线与的距离; ECPD(?)二面角的大小 EPCD,解:(?)以为原点，、分别为 DCDADPD轴建立空间直角坐标系 xyz,由已知可得 DPC(0,0,0),(0,0,2),(0,2,0)设 A(x,0,0)(x,0),则B(x,2,0),113PE,CE得PE,CE,0 由， E(x,0),PE,(x,2),CE,(x,0).2223331332即 由， x,0,故x,.DE,CE,(,0),(,0),0得DE,CE422222又，故是异面直线与的公垂线，
14、易得，故异面直线 CE|DE|,1PDDE,DEPD,的距离为 CEPD1DGPC,(?)作，可设 由DG,PC,0得 Gyz(0,)(0,y,z),(0,2,2),0EFPC,即作于，设， z,2y,故可取DG,(0,1,2),FFmn(0,),31则 EFmn,(,).22,31由， EFPCmnmn,0(,)(0,2,2)0,2120得即22,22312PC又由nmmnEF,，,2,1,(,).故在上得 F22222,EFDG与EPCD,因故的平面角的大小为向量的夹角 EFPCDGPC,DGEF,2,EPCD,.cos,故 即二面角的大小为 ,424|DGEF高考学习网,中国最大高考学习
15、网站G | 我们负责传递知识 ,英文版 , easily blame, to prevent the broken window effect. Supervise the leading cadres to play an exemplary role, take the lead in the strict implementation of the and , lead to safeguard the solemnity and authority of the party discipline, ensure that the party discipline and the law
16、s and regulations for implementation in place. Throughout the discipline in the daily supervision and management , strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Strengthen to key areas, key departments and key projects as well as the mas
17、ses reflect the concentration of the units and departments for supervision. - strengthening supervision, discipline inspection and supervision of cadres to set an example for compliance with the and is a man must be hexyl, blacksmith needs its own hardware. Discipline inspection organs as the execut
18、or of the party discipline, and supervisor of the defenders, for its supervision must be more strictly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of discipline and rules consciousness, politics loyalty, sense obey. Action speak Ji
19、Ordinance to set an example of the 高考学习网,中国最大高考学习网站G | 我们负责传递知识 regulations of the rule of law, strengthen supervision and accept the supervision of the firmness and consciousness, do comply with and . To firmly establish the discipline must first be disciplined, the supervisor will be subject to th
20、e supervision of concept, and consciously safeguard and implement party compasses party, take the lead in practicing three strict real strict, so loyal, clean, play. To be good at learning, the Constitution and the as morality, politics and brought to fruition; to implement , do not want to, dare no
21、t, not with disciplinary ruler to supervision; to discipline a ruler, often the control inspection, and consciously in the ideological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to have quit, the line has ended. Attached: indifferent t
22、o heart, calmly to the table in our life, there are many unpredictable things will happen, some good, some bad things, we cannot control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad things finally turned out to be a good thing, but then we m
23、uddy however did not know, this is the life teach us things. 1, life can be complex, can also be simple. Want simple life of precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to win, 高考学习网,中国最大高考学习网站G | 我们负责传递知识 but the struggle; not
24、to have conquered, but to have fought well. 2, the plain is the background of life. Live a plain life, give up on themselves is not a coward, but the wise answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewarding perfunctorily, but c
25、alm attitude of life of unrestrained self-confidence. Plain living, there is no noise noisy, no earthly troubles, more did not fill in the discontent of desire, some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie, pain is a beginning, the struggle
26、is a kind of process, death is a kind of ending. Give up this giving up is the helpless, do not give up the abandoned, do not give up this giving up is ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is hell. Since the living, to live bet
27、ter. Sometimes we because of too narrow-minded, too care around the chores and penny wise and pound foolish, not worth the candle. Some things to attract trouble and worry, completely depends on how we look at and deal with it. Dont always take everything back to things, and dont get into a blind al
28、ley, dont want to face, dont be narrow-minded. Poke to care, is a kind of open-minded, a free and easy. 5, I am not afraid of others behind me a knife, I afraid to look back and see stab me, is 高考学习网,中国最大高考学习网站G | 我们负责传递知识 my intention to treat people; I am not afraid of the truth to tell the best f
29、riend, Im afraid he turned to it as a joke to tell dont 6, when we are in a positive frame of mind, you will find many good things; and when we are in a negative state of mind, you will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss of sink
30、, Eritrea company. When you are in adversity, may wish to change a point of view to think everything over to the good Think, because good mentality decided the fate of the! 7, people are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a hug. Be
31、cause the world no one can sympathize with you, have mercy on you. You cry, tears is your own; you pain, no one can understand. Then you only tears to smile. 8, each people have youth, Each youth are a story, the life of the world never gets easier, I want what, wish the world all know, as has been
32、the same; now want anything, for fear that others know, or like to lose the same. 9, the heart move, everything in the world is followed by birth, Rangrang, important thing is often the most difficult to open ones mouth, because words will reduce its importance; to let strangers people care about yo
33、ur life in the good things, the original is not easy 10, do not blame, do not laugh at who, also dont envy who. 高考学习网,中国最大高考学习网站G | 我们负责传递知识 Like a person is a kind of feeling, not like a person is true. The truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a stran
34、ge place found a long lost touched. 11, happy life not in the bustling in, and in the peace of mind; no matter how many grievances, how uncomfortable, and ultimately to heal themselves or their own, others may got you to comfort, but never know your heart is how wanjianchuanxin. 12, maam, like a mov
35、ie, learn to appreciate, learn to be grateful, learn tolerance, and goodness, helping others. Instead of accusing the society, as into one; and an exception is better to give than to what 13, dont envy him A sum of, dont lose your life and the life, respectively is: the former is a we experienced ca
36、nnot escape in a day finally will last minute, while the latter is our persistent, we want to cherish the memory of those people and things. 14, learn to smile, learn to strong, the world you know so many people, so many people and you are, you cannot change also cant let everyone like you, so also
37、do not want to do. Life is too short to go crazy to love to go to waste, to chase the dream to regret. 15, when temper, a blessing to go. A wounding elegant people, the key is to control their own emotions. With the mouth is the most stupid behavior. A control negative emotions than a can take a cit
38、y more powerful water flow slow, language is 高考学习网,中国最大高考学习网站G | 我们负责传递知识 expensive. People spent two years of time to learn to speak, but to spend a few years time to shut up. That is a kind of ability, that is a kind of wisdom. 16, life is not perfect, sometimes, growth is not a cry, not an eyeful
39、 of tears, there is no trace of emotion, there is no gleam of hope, no desire, no action, no static, there is only one kind of downward sinking feeling, sink A murky? 6? 7? 6? 7 sink? 6? 7? 6? 7 toward the bottom of the sink. 17, in some way, do not go, you will not know the other side scenery is be
40、autiful. To you is not good, you do not mind too much, no one has an obligation to you; you learn knowledge, is you have weapons, you can start from scratch, but not unarmed; how do you treat people, does not represent how others treat you, if cannot see through this point, only inviting worry. 18,
41、time is like a sponge in the water, as long as you are willing to squeeze, the total water is still there. Every life, after the ups and downs The best test of live, to life, survival and continuation, do not stop the struggle in the joys and sorrows of life on the road, so that different soul to be
42、ar life beat, acceptance of suffering. 19, indifferent to heart, calmly in table, elegant and comfortable life, do not take what is so important. The pursuit will be disappointed; to be alive, you will have trouble. Life is the most afraid of what all want to care about, but also what all grasp is n
43、ot firm, without 高考学习网,中国最大高考学习网站G | 我们负责传递知识 scenery, separated populations, such as not to desire, all docked in the fate of the end. Why is too rigid, the natural, to go stay not to live, let go of obsession, revel is 20, if the fate of the broken Hopes of sailing, please dont despair, the coast
44、is still, if the fate of the withered petals of the beautiful, please do not sink, the spring is still, life will always be endless trouble, please dont helpless, because they are still alive, is still a dream, the sun still, we still. Lost, keep memories; to get, must work to; but the most importan
45、t is good to cherish their own. 21, life, select the complex, is to choose the pain; choose a simple, is choose to be happy. The complex world like a Significance of pride. Hope is the ornate palace, outside people admiring the magnificent, living in the deep knowledge of living it to pay the price.
46、 Simple world as a simple log cabin outside ridiculed shabby, the heart is willing to go live to know the joy. Suffering and joy is their own choice. 22, learn how to use a single powerful heart, let the past be the past, let the future come. Life is really the end of the end of an eagle is flying w
47、ings, life is constantly pursuit. Dont miss to regret, dont wait for old just miss. Time to return, seize every moment, again painstakingly again tired also Those struggling to fly. 23, life could not Yimapingchuan, even flat pavement, inevitably there will be a few pieces of roadblocks. 高考学习网,中国最大高考学习网站G | 我们负责传递知识 Some of the rocks around the past, while others have to move