平面向量讲义(知识点 例题).doc

一、向量的概念与线性运算考点一： 向量及与向量相关的基本概念题型1. 概念判析例1、判断下列各命题是否正确(1)零向量没有方向 (2)若(3)单位向量都相等 (4) 向量就是有向线段(5)两相等向量若共起点,则终点也相同 (6)若，则；(7)若，则 (8)若四边形ABCD是平行四边形,则(9) 的充要条件是且；考点二: 向量的加、减法题型1: 考查加法、减法运算及相关运算律例2、化简题型2: 结合图型考查向量加、减法例3、在所在的平面上有一点，满足，则与的面积之比是( )A B C DABCDE例4、如图，在ABC中，D、E为边AB的两个三等分点，=3a，=2b，求，考点三: 向量数乘运算及其几何意义题型1: 三点共线问题例5、设是不共线的向量，已知向量,若A、B、D三点共线，求k的值。例6、已知A、B、C、P为平面内四点，求证：A、B、C三点在一条直线上的充要条件是存在一对实数m、n，使=m+n，且m+n=1。二、平面向量的基本定理与坐标表示考点一： 平面向量基本定理题型1. 利用一组基底表示平面内的任一向量例7、在OAB中，AD与BC交于点M，设=，=，用,表示。例8、若已知、是平面上的一组基底，则下列各组向量中不能作为基底的一组是 ( )A与 B3与2 C与 D与2例9、在ABC中,已知 AMAB =13, ANAC =14，BN与CM交于点P,且,试 用表示考点二: 平面向量的坐标表示与运算题型1: 向量加、减、数乘的坐标运算例10、已知A（2,4）、B（3,1）、C（3,4）且,求点M、N的坐标及向量的坐标.例11、若A(0, 1), B(1, 2), C(3, 4) 则-2= 例12、若M(3, -2) N(-5, -1) 且 , 求P点的坐标；考点三: 向量平行的充要条件题型1: 平行、共线问题例13、已知向量，若，则锐角等于（ ）A B C D例14、若向量=(-1,x)与=(-x, 2)共线且方向相同，求x例15、已知点O(0，0)，A(1，2)，B(4，5)及，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能否构成为平行四边形？若能，求出相应的t值；若不能，请说明理由三、平面向量的数量积考点一：平面向量数量积的运算题型1. 求数量积、求模、求夹角例16、 ；例17、题型2. 利用数量积解决垂直问题例18、 若非零向量、满足，证明:例19、在ABC中，=(2, 3)，=(1, k)，且ABC的一个内角为直角， 求k值例20、已知向量，若，则（ ） A B C D例21、知为的三个内角的对边，向量若，且，则角的大小分别为（ ）AB C D考点2 利用数量积处理夹角的范围题型1：求夹角范围例22、已知,且关于的方程有实根,则与的夹角的取值范围是 ( )A.0, B. C. D.例23、设非零向量=，=，且，的夹角为钝角，求的取值范围。例24、已知，,如果与的夹角为锐角,则的取值范围是 待添加的隐藏文字内容2Winger 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.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.