高二数学总复习空间向量与立体几何专题讲义(理科版)(20.doc

高二数学总复习选修2-1空间向量与立体几何单元要点归纳讲义（理科版）一、平面法向量的概念和求法 向量与平面垂直：如果表示向量的有向线段所在的直线垂直于平面，则称这个向量垂直于平面，记作。 平面的法向量：如果，那么向量叫做平面的法向量。二. 如何求平面的一个法向量:例题1: 如图1，在正方体ABCD-A1B!C1D1中G、E、F分别为AA1、AB、BC的中点，求平面GEF的法向量。解：以D为原点建立右手空间直角坐标系，则E(1，,0) 、F(，1,0) 、G(1,0，)DABCA1B1C1D1图1GEFxyz由此得: ,设平面的法向量为由及可得令y=1取平面的一个法向量为评析: 因为平面的法向量有无数个，方向可上可下，模可大可小，我们只要求出平面的某一个法向量即可。三、法向量的运用1、利用法向量可处理线面角问题设 为直线与平面所成的角，为直线的方向向量与平面的法向量之间的夹角，则有（图1）或（图2） 图1 图2 特别地 时，；时，或2、 利用法向量可处理二面角问题设 分别为平面的法向量，二面角的大小为，向量 的夹角为，则有（图3）或 （图4）图3 图4例2（北京卷高考题）如图，正三棱柱的底面边长为3，侧棱，D是CB延长线上一点，且。求二面角的大小。（略去了该题的，问）解 取BC的中点O，连AO。由题意 平面平面，平面，以O为原点，建立如图6所示空间直角坐标系， 图7则 ， ， ， ，由题意 平面ABD， 为平面ABD的法向量。设 平面的法向量为 ，则 ， ， ，即 。 不妨设 ，由 ， 得。 故所求二面角的大小为。3 ，利用法向量处理点面距离问题 设 为平面的法向量，A，B分别为平面内，外的点，则点B到平面的距离 略证： 图 例3（全国高考题）如图，已知正四棱柱，点E为中点，点F为中点。求点到平面BDE的距离。（略去了该题的问）考题精选 1，设平面内两个向量的坐标分别为（1，2，1）、（-1，1，2），则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. 已知均为单位向量,它们的夹角为60°,那么等于（ ）A B C D43如图：在平行六面体中，为与的交点。若，则下列向量中与相等的向量是（ ） 4若向量，且与的夹角余弦为，则等于（ ）A B C或 D或5若A，B，当取最小值时，的值等于（ ）A B C D6（2013年普通高等学校招生统一考试大纲版数学（理）已知正四棱柱中,则与平面所成角的正弦值等于（）ABCD7， 若空间三点A（1，5，-2），B（2，4，1），C（p,3,q+2）共线，则p=_,q=_。8若，且，则与的夹角为_。9已知空间四边形，点分别为的中点，且，用，表示，则=_。10（10分）如右图，已知正四棱柱，点E为的中点、点F 为的中点（1）证明为与的公垂线； （2）求到平面的距离。11，如图，已知点P在正方体ABCDA1B1C1D1的ABCDPxyzH对角线BD1上，PDA=60°.（1）求DP与CC1所成角的大小；（2）求DP与平面AA1D1D所成角的大小.待添加的隐藏文字内容312（2013福建理科19）（本小题满分13分）如图，在四棱柱中，侧棱，（1）求证：（2）若直线与平面所成角的正弦值为，求的值；(13（2013年普通高等学校招生全国统一招生考试江苏卷（数学）本小题满分10分.如图,在直三棱柱中,点是的中点(1)求异面直线与所成角的余弦值(2)求平面与所成二面角的正弦值.14.三棱锥被平行于底面的平面所截得的几何体如图所示，截面为，平面，A1AC1B1BDC（）证明：平面平面；（）求二面角的大小（只求余弦值的大小）Winger 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% records heading into the encounter but Australia have not conceded a try since Josh Charnley's effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competition's top try scorer with nine, closely followed by Tuivasa-Sheck with eight.But it is recently named 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 triumphing with the All Blacks in 2011."I'd 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 Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's 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 it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's complaint. In a statement the 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 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 even 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 will 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 - the last try they conceded was scored by England's 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 tough, even contests - strong defence, especially on their own goal-line, completing sets and a good kick-chase. They've 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 into Saturday.The Kiwis are a lot less experienced but winning a dramatic match like their semi-final against England will do wonders for their confidence.They defeated Australia in the Four Nations final in 2010 and the last World Cup, and know they can rise to the big occasion.