高二數(shù)學(xué)總復(fù)習(xí)空間向量與立體幾何專題講義(理科版)(20

1、高二數(shù)學(xué)總復(fù)習(xí)選修2-1空間向量與立體幾何單元要點歸納講義（理科版）一、平面法向量的概念和求法 向量與平面垂直：如果表示向量的有向線段所在的直線垂直于平面，則稱這個向量垂直于平面，記作。 平面的法向量：如果，那么向量叫做平面的法向量。二. 如何求平面的一個法向量:例題1: 如圖1，在正方體abcd-a1b!c1d1中g(shù)、e、f分別為aa1、ab、bc的中點，求平面gef的法向量。解：以d為原點建立右手空間直角坐標系，則e(1，,0) 、f(，1,0) 、g(1,0，)dabca1b1c1d1圖1gefxyz由此得: ,設(shè)平面的法向量為由及可得令y=1取平面的一個法向量為評析: 因為平面的法向量

2、有無數(shù)個，方向可上可下，?？纱罂尚?，我們只要求出平面的某一個法向量即可。三、法向量的運用1、利用法向量可處理線面角問題設(shè) 為直線與平面所成的角，為直線的方向向量與平面的法向量之間的夾角，則有（圖1）或（圖2） 圖1 圖2 特別地 時，；時，或2、 利用法向量可處理二面角問題設(shè) 分別為平面的法向量，二面角的大小為，向量 的夾角為，則有（圖3）或 （圖4）圖3 圖4例2（北京卷高考題）如圖，正三棱柱的底面邊長為3，側(cè)棱，d是cb延長線上一點，且。求二面角的大小。（略去了該題的，問）解 取bc的中點o，連ao。由題意 平面平面，平面，以o為原點，建立如圖6所示空間直角坐標系， 圖7則 ， ， ， ，

3、由題意 平面abd， 為平面abd的法向量。設(shè) 平面的法向量為 ，則 ， ， ，即 。 不妨設(shè) ，由 ， 得。 故所求二面角的大小為。3 ，利用法向量處理點面距離問題 設(shè) 為平面的法向量，a，b分別為平面內(nèi)，外的點，則點b到平面的距離 略證： 圖 例3（全國高考題）如圖，已知正四棱柱，點e為中點，點f為中點。求點到平面bde的距離。（略去了該題的問）考題精選 1，設(shè)平面內(nèi)兩個向量的坐標分別為（1，2，1）、（-1，1，2），則下列向量中是平面的法向量的是（ ） a（-1，-2，5） b（-1,1,-1） c（1, 1,1） d（1，-1，-1）2. 已知均為單位向量,它們的夾角為60,那么等于

4、（ ）a b c d43如圖：在平行六面體中，為與的交點。若，則下列向量中與相等的向量是（ ） 4若向量，且與的夾角余弦為，則等于（ ）a b c或 d或5若a，b，當取最小值時，的值等于（ ）a b c d6（2013年普通高等學(xué)校招生統(tǒng)一考試大綱版數(shù)學(xué)（理）已知正四棱柱中,則與平面所成角的正弦值等于（）abcd7， 若空間三點a（1，5，-2），b（2，4，1），c（p,3,q+2）共線，則p=_,q=_。8若，且，則與的夾角為_。9已知空間四邊形，點分別為的中點，且，用，表示，則=_。10（10分）如右圖，已知正四棱柱，點e為的中點、點f 為的中點（1）證明為與的公垂線； （2）求到平面

5、的距離。11，如圖，已知點p在正方體abcda1b1c1d1的abcdpxyzh對角線bd1上，pda=60.（1）求dp與cc1所成角的大?。唬?）求dp與平面aa1d1d所成角的大小.待添加的隱藏文字內(nèi)容312（2013福建理科19）（本小題滿分13分）如圖，在四棱柱中，側(cè)棱，（1）求證：（2）若直線與平面所成角的正弦值為，求的值；(13（2013年普通高等學(xué)校招生全國統(tǒng)一招生考試江蘇卷（數(shù)學(xué)）本小題滿分10分.如圖,在直三棱柱中,點是的中點(1)求異面直線與所成角的余弦值(2)求平面與所成二面角的正弦值.14.三棱錐被平行于底面的平面所截得的幾何體如圖所示，截面為，平面，a1ac1b1b

6、dc（）證明：平面平面；（）求二面角的大?。ㄖ磺笥嘞抑档拇笮。﹚inger tuivasa-sheck, who scored two tries in the kiwis 20-18 semi-final win over england, has been passed fit after a lower-leg injury, while slater has been named at full-back but is still recovering from a knee injury aggravated against usa.both sides boast 100% rec

7、ords heading into the encounter but australia have not conceded a try since josh charnleys effort in their first pool match against england on the opening day.aussie winger jarryd hayne is the competitions top try scorer with nine, closely followed by tuivasa-sheck with eight.but it is recently name

8、d rugby league international federation player of the year sonny bill williams who has attracted the most interest in the tournament so far.the kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the world cup in both rugby league and rugby union after triu

9、mphing with the all blacks in 2011.id give every award back in a heartbeat just to get across the line this weekend, said williams.the (lack of) air up there watch mcayman islands-based webb, the head of fifas anti-racism taskforce, is in london for the football associations 150th anniversary celebr

10、ations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorrow and i have asked to meet yaya toure, he told bbc sport.for me its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened disciplinary pro

11、ceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body, has also ordered an immediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in a statement the

12、 russian side added: we found no racist insults from fans of cska. baumgartner the disappointing news: mission aborted.the supersonic descent could happen as early as sunda.the weather plays an important role in this mission. starting at the ground, conditions have to be very calm - winds less than

13、2 mph, with no precipitation or humidity and limited cloud cover. the balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. it will climb higher than the tip of mount everest (5.5 miles/8.85 kilometers), drifting eve

14、n higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. as he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.the balloon will slowly drift to the edge of space at 120,000 feet ( then, i would assume, he w

15、ill slowly step out onto something resembling an olympic diving platform.they blew it in 2008 when they got caught cold in the final and they will not make the same mistake against the kiwis in manchester.five years ago they cruised through to the final and so far history has repeated itself here -

16、the last try they conceded was scored by englands josh charnley in the opening game of the tournament.that could be classed as a weakness, a team under-cooked - but i have been impressed by the kangaroos focus in their games since then.they have been concentrating on the sort of stuff that wins you

17、tough, even contests - strong defence, especially on their own goal-line, completing sets and a good kick-chase. theyve been great at all the unglamorous stuff that often goes unnoticed in the stands but not by your team-mates.it is as though their entire tournament has been preparation for the final.in johnathan thurston, cooper cronk, cameron smith and either billy slater or greg inglis at full-back they have a spine that is unmatched in rugby league. they have played in so many high-pressure games - a priceless asset going