(完整word版)高中数学必修4平面向量知识点总结与典型例题归纳(word文档良心出品).pdf

1 平面向量【基本概念与公式】【任何时候写向量时都要带箭头】1.向量：既有大小又有方向的量。记作：AB或a。2.向量的模：向量的大小（或长度），记作：|AB或|a。3.单位向量：长度为 1 的向量。若e是单位向量，则|1e。4.零向量：长度为0 的向量。记作：0。【0方向是任意的，且与任意向量平行】5.平行向量（共线向量）：方向相同或相反的向量。6.相等向量：长度和方向都相同的向量。7.相反向量：长度相等，方向相反的向量。ABBA。8.三角形法则：ABBCAC；ABBCCDDEAE；ABACCB（指向被减数）9.平行四边形法则：以,a b为临边的平行四边形的两条对角线分别为ab，ab。10.共线定理：/abab。当0时，ab与同向；当0时，ab与反向。11.基底：任意不共线的两个向量称为一组基底。12.向量的模：若(,)ax y，则22|axy，22|aa，2|()abab13.数量积与夹角公式：|cosa bab；cos|a bab14.平行与垂直：1221/ababx yx y；121200aba bx xy y题型 1.基本概念判断正误：（1）共线向量就是在同一条直线上的向量。（2）若两个向量不相等，则它们的终点不可能是同一点。（3）与已知向量共线的单位向量是唯一的。（4）四边形 ABCD 是平行四边形的条件是ABCD。（5）若ABCD，则 A、B、C、D四点构成平行四边形。（6）若a与b共线，b与c共线，则a与c共线。（7）若mamb，则ab。2（8）若mana，则mn。（9）若a与b不共线，则a与b都不是零向量。（10）若|a bab，则/ab。（11）若|abab，则ab。题型 2.向量的加减运算1.设a表示“向东走8km”,b表示“向北走6km”,则|ab。2.化简()()ABMBBOBCOM。3.已知|5OA,|3OB,则|AB的最大值和最小值分别为、。4.已知ACABAD为与的和向量，且,ACa BDb，则AB，AD。5.已知点 C在线段 AB上，且35ACAB,则ACBC，ABBC。题型 3.向量的数乘运算1.计算：2(253)3(232)abcabc2.已知(1,4),(3,8)ab，则132ab。题型 4.根据图形由已知向量求未知向量1.已知在ABC中，D是BC的中点，请用向量AB AC，表示AD。2.在平行四边形ABCD中，已知,ACa BDb，求ABAD和。题型 5.向量的坐标运算1.已知(4,5)AB，(2,3)A，则点B的坐标是。2.已知(3,5)PQ，(3,7)P，则点Q的坐标是。3.若物体受三个力1(1,2)F,2(2,3)F,3(1,4)F,则合力的坐标为。4.已知(3,4)a，(5,2)b，求ab，ab，32ab。5.已知(1,2),(3,2)AB,向量(2,32)axxy与AB相等，求,x y的值。文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 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HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L63 6.已知(2,3)AB，(,)BCm n，(1,4)CD，则DA。7.已知O是坐标原点，(2,1),(4,8)AB，且30ABBC，求OC的坐标。题型 6.判断两个向量能否作为一组基底1.已知12,e e是平面内的一组基底，判断下列每组向量是否能构成一组基底：A.1212eeee和 B.1221326eeee和4 C.122133eeee和 D.221eee和2.已知(3,4)a，能与a构成基底的是（）A.3 4(,)5 5 B.4 3(,)5 5 C.34(,)55 D.4(1,)3题型 7.结合三角函数求向量坐标1.已知O是坐标原点，点A在第二象限，|2OA，150 xOA，求OA的坐标。2.已知O是原点，点A在第一象限，|4 3OA，60 xOA，求OA的坐标。题型 8.求数量积1.已知|3,|4ab，且a与b的夹角为60，求（1）a b，（2）()aab，（3）1()2abb，（4）(2)(3)abab。2.已知(2,6),(8,10)ab，求（1）|,|ab，（2）a b，（3）(2)aab，（4）(2)(3)abab。题型 9.求向量的夹角1.已知|8,|3ab，12a b，求a与b的夹角。文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 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11.求单位向量【与a平行的单位向量：|aea】1.与(12,5)a平行的单位向量是 2.与1(1,)2m平行的单位向量是。题型 12.向量的平行与垂直1.已知(1,2)a，(3,2)b，（1）k为何值时，向量kab与3ab垂直？（2）k为何值时向量kab与3ab平行？2.已知a是非零向量，a ba c，且bc，求证：()abc。题型 13.三点共线问题文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 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ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L65 1.已知(0,2)A，(2,2)B，(3,4)C，求证：,A B C三点共线。2.设2(5),28,3()2ABabBCab CDab，求证：ABD、三点共线。3.已知2,56,72ABab BCab CDab，则一定共线的三点是。4.已知(1,3)A，(8,1)B，若点(21,2)Caa在直线AB上，求a的值。5.已知四个点的坐标(0,0)O，(3,4)A，(1,2)B，(1,1)C，是否存在常数t，使O A t O BO C成立？题型 14.判断多边形的形状1.若3ABe，5CDe，且|ADBC,则四边形的形状是。2.已知(1,0)A，(4,3)B，(2,4)C，(0,2)D，证明四边形ABCD是梯形。3.已知(2,1)A，(6,3)B，(0,5)C，求证：ABC是直角三角形。4.在平面直角坐标系内，(1,8),(4,1),(1,3)OAOBOC,求证：ABC是等腰直角三角形。题型 15.平面向量的综合应用1.已知(1,0)a，(2,1)b，当k为何值时，向量kab与3ab平行？文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L66 2.已知(3,5)a，且ab，|2b，求b的坐标。3.已知ab与同向，(1,2)b，则10a b，求a的坐标。4.已知(1,2)a，(3,1)b，(5,4)c，则cab。5.已知(,3)am，(2,1)b，（1）若a与b的夹角为钝角，求m的范围；（2）若a与b的夹角为锐角，求m的范围。6.已知(6,2)a，(3,)bm，当m为何值时，（1）a与b的夹角为钝角？（2）a与b的夹角为锐角？7.已知梯形ABCD的顶点坐标分别为(1,2)A，(3,4)B，(2,1)D，且/ABDC，2ABCD，求点C的坐标？8.已知ABC三个顶点的坐标分别为(3,4)A，(0,0)B，(,0)C c，（1）若0AB AC，求c的值；（2）若5c，求sin A的值？文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 ZZ3F6N10F9L6文档编码:CS8H10P3M7T3 HK9H9J3S9Y3 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