第五章平面向量、復(fù)數(shù)[知識(shí)網(wǎng)絡(luò)][命題方向]1．高考對平面向量與復(fù)數(shù)的考查較為穩(wěn)定，都是一個(gè)選擇或填空題，復(fù)數(shù)試題為容易題，向量的難度為中檔或偏下．(1)在高考試題中以平面向量的線性運(yùn)算、平面向量的數(shù)量積為主要考查點(diǎn)，考查學(xué)生對平面向量的基本知識(shí)和基本解題方法的掌握情況．(2)在處理與平面向量有關(guān)的問題時(shí)，注意坐標(biāo)法和基底法在解題中的應(yīng)用．重點(diǎn)考查的學(xué)科核心素養(yǎng)為數(shù)學(xué)運(yùn)算和邏輯推理．2．向量的考查內(nèi)容為：(1)向量的概念與線性運(yùn)算，(2)以平面向量基本定理為核心的向量的坐標(biāo)表示、向量的坐標(biāo)運(yùn)算，(3)以平面向量的數(shù)量積運(yùn)算為載體，考查向量的夾角、模以及垂直的充要條件的應(yīng)用．3．復(fù)數(shù)考查的內(nèi)容主要集中在：(1)復(fù)數(shù)的相關(guān)概念、幾何意義，(2)復(fù)數(shù)的四則運(yùn)算，一般為除法運(yùn)算．探究1(人教A版必修第二冊P8)向量加法的平行四邊形法則與三角形法則一致嗎？為什么？_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版必修第二冊P9)(1)如果向量a，b共線，它們的加法與數(shù)的加法有什么關(guān)系？你能作出向量a＋b嗎？(2)結(jié)合例1，探索|a＋b|，|a|，|b|之間的關(guān)系．_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究3(人教A版必修第二冊P33)如圖，線段P1P2的端點(diǎn)P1，P2的坐標(biāo)分別是(x1，y1)，(x2，y2)，點(diǎn)P是直線P1P2上的一點(diǎn)．當(dāng)eq\o(P1P,\s\up6(→))＝λeq\o(PP2,\s\up6(→))時(shí)，點(diǎn)P的坐標(biāo)是什么？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究4(人教A版必修第二冊P69)復(fù)數(shù)集C與實(shí)數(shù)集R之間有什么關(guān)系？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究5(人教A版必修第二冊P72)若z1，z2是共軛復(fù)數(shù)，那么在復(fù)平面內(nèi)它們所對應(yīng)的點(diǎn)有怎樣的關(guān)系？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題1(人教A版必修第二冊P14例6)如圖，?ABCD的兩條對角線相交于點(diǎn)M，且eq\o(AB,\s\up6(→))＝a，eq\o(AD,\s\up6(→))＝b，用a，b表示eq\o(MA,\s\up6(→))，eq\o(MB,\s\up6(→))，eq\o(MC,\s\up6(→))和eq\o(MD,\s\up6(→)).____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·新高考Ⅰ卷)在△ABC中，點(diǎn)D在邊AB上，BD＝2DA.記eq\o(CA,\s\up6(→))＝m，eq\o(CD,\s\up6(→))＝n，則eq\o(CB,\s\up6(→))＝()A．3m－2n B．－2m＋3nC．3m＋2n D．2m＋3n點(diǎn)評(píng)本題與教材例題都是考查向量的線性運(yùn)算．典題2(人教A版必修第二冊P16例8)已知a，b是兩個(gè)不共線的向量，向量b－ta，eq\f(1,2)a－eq\f(3,2)b共線，求實(shí)數(shù)t的值．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題3(人教A版必修第二冊P21例12)已知|a|＝6，|b|＝4，a與b的夾角為60°，求(a＋2b)·(a－3b)．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題4(人教A版必修第二冊P21例13)已知|a|＝3，|b|＝4，且a與b不共線．當(dāng)k為何值時(shí)，向量a＋kb與a－kb互相垂直？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題5(人教A版必修第二冊P22練習(xí)T3)求證：(a＋b)2－(a－b)2＝4a·b.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題6(人教A版必修第二冊P24習(xí)題6.2T18)已知|a|＝4，|b|＝3，且(2a－3b)·(2a＋b)＝61，求a與b的夾角θ.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·全國甲卷)已知向量a，b，c滿足|a|＝|b|＝1，|c|＝eq\r(2)，且a＋b＋c＝0，則cos〈a－c，b－c〉＝()A．－eq\f(4,5) B．－eq\f(2,5)C.eq\f(2,5) D.eq\f(4,5)點(diǎn)評(píng)本題與教材習(xí)題都是考查利用數(shù)量積公式求夾角．典題7(人教A版必修第二冊P24習(xí)題6.2T21)已知△ABC的外接圓圓心為O，且2eq\o(AO,\s\up6(→))＝eq\o(AB,\s\up6(→))＋eq\o(AC,\s\up6(→))，|eq\o(OA,\s\up6(→))|＝|eq\o(AB,\s\up6(→))|，則向量eq\o(BA,\s\up6(→))在向量eq\o(BC,\s\up6(→))上的投影向量為()A.eq\f(1,4)eq\o(BC,\s\up6(→)) B.eq\f(\r(3),4)eq\o(BC,\s\up6(→))C．－eq\f(1,4)eq\o(BC,\s\up6(→)) D．－eq\f(\r(3),4)eq\o(BC,\s\up6(→))_______________________________________________________________________________________________________________________________________________________________________________________________________________典題8(人教A版必修第二冊P24習(xí)題6.2T22)如圖，O是平行四邊形ABCD外一點(diǎn)，用eq\o(OA,\s\up6(→))，eq\o(OB,\s\up6(→))，eq\o(OC,\s\up6(→))表示eq\o(OD,\s\up6(→)).____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題9(人教A版必修第二冊P24習(xí)題6.2T23)已知O為四邊形ABCD所在平面內(nèi)一點(diǎn)，且向量eq\o(OA,\s\up6(→))，eq\o(OB,\s\up6(→))，eq\o(OC,\s\up6(→))，eq\o(OD,\s\up6(→))滿足等式eq\o(OA,\s\up6(→))＋eq\o(OC,\s\up6(→))＝eq\o(OB,\s\up6(→))＋eq\o(OD,\s\up6(→)).(1)作出滿足條件的四邊形ABCD.(2)四邊形ABCD有什么特點(diǎn)？請證明你的猜想．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版必修第二冊P24習(xí)題6.2T24)如圖，在⊙C中，是不是只需知道⊙C的半徑或弦AB的長度，就可以求出eq\o(AB,\s\up6(→))·eq\o(AC,\s\up6(→))的值？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題11(人教A版必修第二冊P26例1)如圖，eq\o(OA,\s\up6(→))，eq\o(OB,\s\up6(→))不共線，且eq\o(AP,\s\up6(→))＝teq\o(AB,\s\up6(→))(t∈R)，用eq\o(OA,\s\up6(→))，eq\o(OB,\s\up6(→))表示eq\o(OP,\s\up6(→)).觀察eq\o(OP,\s\up6(→))＝(1－t)eq\o(OA,\s\up6(→))＋teq\o(OB,\s\up6(→))，你有什么發(fā)現(xiàn)？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題12(人教A版必修第二冊P30例5)如圖，已知?ABCD的三個(gè)頂點(diǎn)A，B，C的坐標(biāo)分別是(－2，1)，(－1，3)，(3，4)，求頂點(diǎn)D的坐標(biāo)．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題13(人教A版必修第二冊P31例7)已知a＝(4，2)，b＝(6，y)，且a∥b，求y.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·全國乙卷)已知向量a＝(2，5)，b＝(λ，4)，若a∥b，則λ＝________．典題14(人教A版必修第二冊P37習(xí)題6.3T15)如圖，設(shè)Ox，Oy是平面內(nèi)相交成60°角的兩條數(shù)軸，e1，e2分別是與x軸、y軸正方向同向的單位向量．若向量eq\o(OP,\s\up6(→))＝xe1＋ye2，則把有序數(shù)對(x，y)叫做向量eq\o(OP,\s\up6(→))在坐標(biāo)系xOy中的坐標(biāo)．設(shè)eq\o(OP,\s\up6(→))＝3e1＋2e2.(1)計(jì)算|eq\o(OP,\s\up6(→))|的大小；(2)根據(jù)平面向量基本定理判斷，本題中對向量坐標(biāo)的規(guī)定是否合理．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題15(人教A版必修第二冊P37習(xí)題6.3T16)用向量方法證明：對于任意的a，b，c，d∈R，恒有不等式(ac＋bd)2≤(a2＋b2)(c2＋d2)．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題16(人教A版必修第二冊P40例3)在日常生活中，我們有這樣的經(jīng)驗(yàn)：兩個(gè)人共提一個(gè)旅行包，兩個(gè)拉力夾角越大越費(fèi)力；在單杠上做引體向上運(yùn)動(dòng)，兩臂的夾角越小越省力，你能從數(shù)學(xué)的角度解釋這種現(xiàn)象嗎？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題17(人教A版必修第二冊P52習(xí)題6.4T1)若非零向量eq\o(AB,\s\up6(→))與eq\o(AC,\s\up6(→))滿足eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(\o(AB,\s\up6(→)),|\o(AB,\s\up6(→))|)＋\f(\o(AC,\s\up6(→)),|\o(AC,\s\up6(→))|)))·eq\o(BC,\s\up6(→))＝0，且eq\f(\o(AB,\s\up6(→)),|\o(AB,\s\up6(→))|)·eq\f(\o(AC,\s\up6(→)),|\o(AC,\s\up6(→))|)＝eq\f(1,2)，則△ABC為()A．三邊均不相等的三角形 B．直角三角形C．底邊和腰不相等的等腰三角形 D．等邊三角形_______________________________________________________________________________________________________________________________________________________________________________________________________________典題18(人教A版必修第二冊P52習(xí)題6.4T2)已知O，N，P在△ABC所在平面內(nèi)，滿足|eq\o(OA,\s\up6(→))|＝|eq\o(OB,\s\up6(→))|＝|eq\o(OC,\s\up6(→))|，eq\o(NA,\s\up6(→))＋eq\o(NB,\s\up6(→))＋eq\o(NC,\s\up6(→))＝0，且eq\o(PA,\s\up6(→))·eq\o(PB,\s\up6(→))＝eq\o(PB,\s\up6(→))·eq\o(PC,\s\up6(→))＝eq\o(PC,\s\up6(→))·eq\o(PA,\s\up6(→))，則點(diǎn)O，N，P依次是△ABC的()A．重心，外心，垂心 B．重心，外心，內(nèi)心C．外心，重心，垂心 D．外心，重心，內(nèi)心_______________________________________________________________________________________________________________________________________________________________________________________________________________典題19(人教A版必修第二冊P60復(fù)習(xí)參考題6T8)已知向量a＝(1，0)，b＝(1，1)．當(dāng)λ為何值時(shí)，a＋λb與a垂直？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·新高考Ⅰ卷)已知向量a＝(1，1)，b＝(1，－1)．若(a＋λb)⊥(a＋μb)，則()A．λ＋μ＝1 B．λ＋μ＝－1C．λμ＝1 D．λμ＝－1點(diǎn)評(píng)教材習(xí)題和高考題類似，都是根據(jù)向量垂直的充要條件求參數(shù)，其一般方法為根據(jù)平面向量數(shù)量積的坐標(biāo)表示以及向量的線性運(yùn)算列出方程，即可解出．典題20(人教A版必修第二冊P61復(fù)習(xí)參考題6T13(5))已知等邊三角形ABC的邊長為1，eq\o(BC,\s\up6(→))＝a，eq\o(CA,\s\up6(→))＝b，eq\o(AB,\s\up6(→))＝c，那么a·b＋b·c＋c·a＝()A．3 B．－3C.eq\f(3,2) D．－eq\f(3,2)_______________________________________________________________________________________________________________________________________________________________________________________________________________(2022·全國乙卷)已知向量a，b滿足|a|＝1，|b|＝eq\r(3)，|a－2b|＝3，則a·b＝()A．－2 B．－1C．1 D．2點(diǎn)評(píng)本題和教材習(xí)題考查的都是數(shù)量積的基本運(yùn)算，并且在形式上完全相同，不同之處是高考試題沒有給出各向量的夾角，需要通過平方進(jìn)行轉(zhuǎn)化．而事實(shí)上，教材習(xí)題中的條件也是滿足a＋b＋c＝0，也就是說教材習(xí)題的解法可以按照高考題的解法進(jìn)行，看起來，在學(xué)習(xí)教材時(shí)，對習(xí)題進(jìn)行深刻的挖掘還是很有必要的．典題21(人教A版必修第二冊P61復(fù)習(xí)參考題6T13(6))若平面向量a，b，c兩兩的夾角相等，且|a|＝1，|b|＝1，|c|＝3，則|a＋b＋c|＝()A．2 B．5C．2或5 D.eq\r(2)或eq\r(5)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·新高考Ⅱ卷)已知向量a，b滿足|a－b|＝eq\r(3)，|a＋b|＝|2a－b|，則|b|＝_________________.點(diǎn)評(píng)高考題和教材習(xí)題考查的都是求向量的模，一般方法就是根據(jù)題目條件，利用向量模的平方得出答案．典題22(人教A版必修第二冊P61復(fù)習(xí)參考題6T14)已知a，b，c，d為非零向量，證明下列結(jié)論，并解釋其幾何意義．(1)a⊥b?|a＋b|＝|a－b|；(2)若a＋b＝c，a－b＝d，則|a|＝|b|?c⊥d.______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題23(人教A版必修第二冊P69例1)當(dāng)實(shí)數(shù)m取什么值時(shí)，復(fù)數(shù)z＝m＋1＋(m－1)i是下列數(shù)？(1)實(shí)數(shù)；(2)虛數(shù)；(3)純虛數(shù)．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題24(人教A版必修第二冊P73習(xí)題7.1T3)求適合下列方程的實(shí)數(shù)x與y的值：(1)(3x＋2y)＋(5x－y)i＝17－2i；(2)(x＋y－3)＋(x－4)i＝0.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·全國乙卷)設(shè)(1＋2i)a＋b＝2i，其中a，b為實(shí)數(shù)，則()A．a(chǎn)＝1，b＝－1 B．a(chǎn)＝1，b＝1C．a(chǎn)＝－1，b＝1 D．a(chǎn)＝－1，b＝－1點(diǎn)評(píng)本題和教材習(xí)題考查角度完全一致，都是利用復(fù)數(shù)相等的條件求參數(shù)值，且其難度小于教材習(xí)題的難度．典題25(人教A版必修第二冊P79例5)計(jì)算(1＋2i)÷(3－4i)．________________________________________________________________________________________________________________________________________________________________________________________________