平面向量講義(知識(shí)點(diǎn) 例題)

1、專業(yè)好文檔一、向量的概念與線性運(yùn)算考點(diǎn)一： 向量及與向量相關(guān)的基本概念題型1. 概念判析例1、判斷下列各命題是否正確(1)零向量沒有方向 (2)若(3)單位向量都相等 (4) 向量就是有向線段(5)兩相等向量若共起點(diǎn),則終點(diǎn)也相同 (6)若，則；(7)若，則 (8)若四邊形abcd是平行四邊形,則(9) 的充要條件是且；考點(diǎn)二: 向量的加、減法題型1: 考查加法、減法運(yùn)算及相關(guān)運(yùn)算律例2、化簡題型2: 結(jié)合圖型考查向量加、減法例3、在所在的平面上有一點(diǎn)，滿足，則與的面積之比是( )a b c dabcde例4、如圖，在abc中，d、e為邊ab的兩個(gè)三等分點(diǎn)，=3a，=2b，求，考點(diǎn)三: 向量數(shù)

2、乘運(yùn)算及其幾何意義題型1: 三點(diǎn)共線問題例5、設(shè)是不共線的向量，已知向量,若a、b、d三點(diǎn)共線，求k的值。例6、已知a、b、c、p為平面內(nèi)四點(diǎn)，求證：a、b、c三點(diǎn)在一條直線上的充要條件是存在一對(duì)實(shí)數(shù)m、n，使=m+n，且m+n=1。二、平面向量的基本定理與坐標(biāo)表示考點(diǎn)一： 平面向量基本定理題型1. 利用一組基底表示平面內(nèi)的任一向量例7、在oab中，ad與bc交于點(diǎn)m，設(shè)=，=，用,表示。例8、若已知、是平面上的一組基底，則下列各組向量中不能作為基底的一組是 ( )a與 b3與2 c與 d與2例9、在abc中,已知 amab =13, anac =14，bn與cm交于點(diǎn)p,且,試 用表示考點(diǎn)二

3、: 平面向量的坐標(biāo)表示與運(yùn)算題型1: 向量加、減、數(shù)乘的坐標(biāo)運(yùn)算例10、已知a（2,4）、b（3,1）、c（3,4）且,求點(diǎn)m、n的坐標(biāo)及向量的坐標(biāo).例11、若a(0, 1), b(1, 2), c(3, 4) 則-2= 例12、若m(3, -2) n(-5, -1) 且 , 求p點(diǎn)的坐標(biāo)；考點(diǎn)三: 向量平行的充要條件題型1: 平行、共線問題例13、已知向量，若，則銳角等于（ ）a b c d例14、若向量=(-1,x)與=(-x, 2)共線且方向相同，求x例15、已知點(diǎn)o(0，0)，a(1，2)，b(4，5)及，求(1)t為何值時(shí)，p在x軸上？p在y軸上？p在第二象限。 (2)四邊形oabp

4、能否構(gòu)成為平行四邊形？若能，求出相應(yīng)的t值；若不能，請(qǐng)說明理由三、平面向量的數(shù)量積考點(diǎn)一：平面向量數(shù)量積的運(yùn)算題型1. 求數(shù)量積、求模、求夾角例16、 ；例17、題型2. 利用數(shù)量積解決垂直問題例18、 若非零向量、滿足，證明:例19、在abc中，=(2, 3)，=(1, k)，且abc的一個(gè)內(nèi)角為直角， 求k值例20、已知向量，若，則（ ） a b c d例21、知為的三個(gè)內(nèi)角的對(duì)邊，向量若，且，則角的大小分別為（ ）ab c d考點(diǎn)2 利用數(shù)量積處理夾角的范圍題型1：求夾角范圍例22、已知,且關(guān)于的方程有實(shí)根,則與的夾角的取值范圍是 ( )a.0, b. c. d.例23、設(shè)非零向量=，=

5、，且，的夾角為鈍角，求的取值范圍。例24、已知，,如果與的夾角為銳角,則的取值范圍是 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 bo

6、ast 100% records 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 r

7、ecently 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 unio

8、n after triumphing 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 annive

9、rsary celebrations 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 disc

10、iplinary proceedings 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 s

11、tatement 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 - wind

12、s 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),

13、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

14、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 it

15、self here - 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 th

16、at wins you 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

17、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 w

18、inning 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

19、 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 charnleys effort in their

20、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 named rugby league international federation player of the year sonny bill williams who has attracted the most i

21、nterest 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.id give every award back in a heartbeat just to get across the line this

22、 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 celebrations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorr

23、ow 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 proceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president o

24、f 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 russian side added: we found no racist insults from fans of cska. baumgartner the disappointing news: miss

25、ion 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

26、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 strato

27、sphere. 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

28、 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 englands josh charnley in the opening game of the tournament.that could be classed as a weakness, a team under-cooked - but i have