数学重难点必刷题：平面向量含详解

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18x<0y>0Å4Í=VÑீåḄ>Ǽ᝞ÕÙÉ▢º×ᑖᡠN,x+jy/஺+AABCḄ:¢ø᪷öMᙠAOBCᑁᑨû4M஺:ᔠ4+2஻ᨬüøMC:ᔠý+2஻ḄcᨬᜧþÍ¿ÀÁᡠ)ÿ
ᓽ.ூ௃஺A4BCᑁ++=6ᡠ0AABCḄ!Mᙠ$OBCᑁ&'()*+,M-0ᔠ/2+2஻ᨬ45/AM=AAB+^AC^^x+=+ᡠ/[=!,஻=’,ᓽ;+2஻=133,M-Cᔠ/2+2஻ᨬᜧ5/AM=ACᡠX=O,4=1,ᓽ4+24=2MᙠNOBCᑁ'()*@ᫀB9⚓D58⚓

19ᡠEF
ᓽG1+246(1,2)ᡠ⌱BூK௃L⚪ὃOPᔣRᙠSTUVḄWឋYZᱯ\]^_Ḅ`ᵨbcd⚪.9.Aூᑖ᪆௃ᐜᵫj=/1k+(஻—1)mn)opqᑮGIḄᐗuvpwᙠpw'Ij)___________________Ae¥+”{ᨵnqᑮ஻ḄE}ᵫVp=+஻m=i,>qᑮ஻-
Ḅᐵ/Ḅ.ூ௃ᵫ□C:(x—2y+(y—2=2,q⊈|m|=2&0T஺ḄᩩᑗWᡠ|p|=75|k"OTCT=QOCOT=\pT^=6,OC-CT=-\CT^=-2=/_¢=/1k+(4—1)mᡠ2=[/1k+£Pu;2k2+2¤(஻_])¦ᐗ+(஻_02§2ᓽ2=6¤2+12/1(஻-1)+8(஻-1)22ªG'ᓄ®q322+6(஻-1)2+4(஻—1)~—1=0,ᙠ¤£{ᨵn□=[6(஻-1)]2-4X3[4(஻-1)2-1>0ᡠ-aoT3+6(஻1)#+4(஻1)212஺q0<஻<1ᑶ@ᫀB10⚓D58⚓

20µOPCT=(/lor+x/OCjcT=AOTCT+/.tOCCT=-2஻=tᡠº-24f<0ᦑ⌱A.ூK௃L⚪ὃOPᔣRḄᦪR¾¿ᐸ`ᵨᐗuvpwÁ᪷ḄᑖÃÄᔠឋÅÆbcd⚪.10.Cூᑖ᪆௃ᐜÇÈᔣR4=(4,0),ᑣᔣRʪᑖËᙠ(4,0)ᙊ!rḄᙊ{ḄÍnÎYÏ,qÐBḄᙶ᪗ᑭᵨOB=BC,qcos®=£uÖᯠØᑭᵨo☢ᔣRḄÚÛÜÝ8rÞ+ß!/?)Ḅᨬ4}%,,4nm=3rsin6,ᯠØq@ᫀᓽ.ூ௃᝞åÇÈᔣRa=(4,0),ᑣᔣR5çᑖËᙠ(4,0)ᙊ!rḄᙊ{ḄÍnÎYÏᐸVè=஺=Ê=3᧕ÐBḄᙶ᪗B(rcos8+4,rsine)MᡠOB=BC,BP(rcos^+4)'+r2sin2^=4r2᦮ᳮqcos©=3/16,ᡠsin®=Jl_cos2e=J^+16°>-2568rV64ñW+ófeR)Ḅᨬ4}4M,1@ᫀB11⚓D58⚓

21pn,a.Aa19/+160—256ᓽ=3rsin஺=3J-------------QQ2÷—9/+160ᡝ256=—9&ñJ+160,256,,ñ=—§/Eᨬᜧ}5/19/+160ᡝ-256<8V64-3ᦑ4☢Ḅᨬᜧ}8ᦑ⌱CூK௃L⚪ú⌕ὃOPo☢ᔣR-o☢ÚÛḄÄᔠÐüᑭᵨᙊḄឋýo☢ᔣRḄÚÛÜÝ⍝ÄᔠឋÅÆḄ⚪ÿ
⚪.11.Dூᑖ᪆௃ᵫ•=ᑮ(+)=0ᑭᵨ☢ᔣ!"Ḅ$%&'᪀⌼*+,-BCD4,᪷4m•(+)=0,6*+,-3s49:-;᪵ᙠ&ABOAABC:-Ḅ,DBC.ᐗḄ⊤GHᑭᵨ;᪀⌼JᦪᨬMBC.NḄOPR.ூST௃BC-CA=CA-AB^>CA-(BC-AB)=Q^CA-(,BC+BA)=O,BC,8AVW,X*+,-BCZM,᝞\]^BrCᡠ`a+=bcd•=0nC4,8஺ᡠ*+,-5CQ49:-ijᫀl12⚓n58⚓

22ULI'UlinC4cBD=0,8A+BC=2,ᡠ=2n8O=l,ᙠR/ABOAA,cos/ABO=ᔞnABABZAcBoCs--------2______12p"2cosZABCy-BABC^^ABCr2-cosZABC=-------------1+cosAABCcos2ix=cosZABC•.♦tWNABCீ4v...Xe[-lw],3322112x222ᡠyx+”zᡝ|᱄=~஺kM9Jᦪᦑy/yj,bc⚪⌱D.ூ௃⚪ὃ☢!"Ḅ$%&'ஹ☢ᔣᦪḄOPR⚪ᑭᵨ:-ḄឋஹḄᓣHஹ᪀⌼Jᦪ9T⚪Ḅᐵ.12.CூT᪆௃3C4,CBVxyᙶ᪗U^44,0r,30,3r,!^y=3-8i
3ஹ38M\a,3--a\,NQb,3--bR,Ꮇia0¶ᓽᓲ;I+3+A2r-l-V2-cos^>0+5a+3>0/£3஺0'A2-3^=0ᐭ2—3Ho'Tae-8,2^1ru^1^2,73ruÃ,+8r.^⚪Ä⌕ὃ☢ᔣḄᦪÆÇÈᜳHὃ┦ÉÊËJᦪḄOPRὃ§ÌᔣḄÍÎᐵ.⚪Ï£ÐÑÒÓ§ÌᔣḄÔÕᜳÖ×ØÙḄᦪÚHBTCᩭ.ᵫM☢ÒÓ§ÌᔣḄᜳV┦ᑣÝᓄVᦪᜧàájᫀl13⚓n58⚓

23âã1,äåæ§Ìᔣâçᐳéᵫcᑮ;12-3H0.14.AூT᪆௃ê⚪ᑖ᪆^᪷4ëᙊḄÓ'ÕNíᨵ|PN|+|QN|=|-(IDF/+|ò|)=|-4=10.ὃ^éÈᙊ┵ôéÍÎᐵ.15.DூT᪆௃ê⚪ᑖ᪆^ᵫ|"|=13,ᓛ|=1,y-5ᑖ12,0£5v÷=169+25—107v4144,᦮ᳮ____ýಘ5IT஺25,᪷ᑣ/,ᙠ஺Ḅᢗᐸᢗᜧ6=1,ᡠ"1311%ᐸ'ᦑ⌱D.13ὃ-.ᔣ0ᙠ123ᔣ0Ḅ4ᔣḄᢗḄ'5⚪.16.Cூ9᪆௃9.V|A|A2|=2a=4,2c=2Bb=l,EP(xo,yo),•MNFIPF2=90°N,SF1PF2=,'1"/2x2V3xyo=lxtan(900/2),A9yo=MXyo=/Yᐭ[ᙊ]+_=1xo=`aᵫPFrPF<0,NF1PF2N90஺.2fᔠ⚪EᩩijklPF).PF<0ḄM-Ḅᭆ᳛=o-(-∅)/2a=a/4ὃ217.BDூᑖ᪆௃VCOS6T=1(4+l)(sina+l)ENB4P=a᪷ᔣ0yz,,{|+y=1.}'1ysma-x=i%sina+cosa/(&)=""»-1«e[0,-],ᦪᦪᦪᨬzsma+cosa2/(a)min=""-I=`1,949ᓽj.m,n22+12ᫀ14⚓58⚓

24ூ9௃EZ.BAP=aᑣAP=cosaAB+sinaADᵫAC=xDE+yAPjAC=x(AE-A஺)+y(cosaAB+sinaAD)=(xA+ycosa)AB+(ysina-x)ADd+ycosa=1=+ᡠ%ysina-x=lsina-cos஺x---------------------Zsin^+cos«92+1y=-------------------asina+cosasin஺2cosa2+1ᡠ%x+y=—-----------------1-----------------------Asincr+cos

2522-1(2+1)(-^—-+1)+2A1y/3+1ᡠ%/(a)min=/(4)=------U-22-l22^22+l+A2+l94ᡈ2+JLᦑ⌱.BD.ூ-°௃ᐵ²--°.ᙠᜐᳮx+y=(/+D.ina+1)_¶ḄᨬN,ᑭᵨᦪ¹º⚪Ḅᐵ²Xsina+cosa-¼¹½-¾¿À¿ÁÂÃᦪḄ᩽-ᓽᨬ-Å9º⚪Ḅyz0Æᜧǽ⚪.18.ACDூᑖ᪆௃᪷᜜ÊᙠABḄ˹ABḄÌ-ᑭᵨᔣ0Ḅᦪ0ÍḄÎj%ÏÐAÑÒÓᑭᵨᔣ0Ḅᦪ0ÍḄyzÔᑣj%Õ9=□×ᓽ04_LBC,ᙠ2ØÁÙÌ᧕kÚ¹2ÑÒḄᵫ£jᑨB┯ÝÓᑭᵨÁÙÌÞḄÎÞឋyzàá☢ᔣ0ãºᳮÌḄäåj%ÏÐCÑÒÓᑭᵨᔣ0Ḅᦪ0Íyzàᔣ0ᚖçḄᩩij%ᑨèé1ᚖçÅêÐDÑÒ.ூ9௃᝞ìEABÌ-M,ᑣOM_LAB,|AacQsNOAV=|AM|AOABcosNOAB=î(ïcosNOAB)=|AB|J^1]_ðAÑ2Ò;ᫀ16⚓58⚓

26OAOB=OAOCòóOA'OB-OC=0òóOA-CB=0,ᓽOABC,ô2ØÁÙõ஺¹᜜Ê2èᚖçö᝞çÁÁÙABCÌ,÷3çÁ⚔-ᑣ஺ùúACḄÌ-èᚖç.ᦑB┯ÝÓEBCḄÌ-஺ᑣûü+ṹ=4þÿ+
=+33ஹ>஻)323஻EEGᐳ'•■+=1,ᓽ35=3,ᦑC;cosC-|BC|+|BC|=O,ABACᓱᡭᚖ'!,#$cos%cosC&ᐳᦑD.ᦑ⌱ACD.ூ*௃,⚪ὃ/0☢ᔣ3ឋ567ᦪ3956ᔣ3ᚖ7ᐳḄᑨ<0☢ᔣ3ᑖ>Ḅ?,<ᳮABᔠD⚪EFGᜧᐵJKLMNᵨᔣ3Ḅឋ567ᦪ3P56ᳮ>ᐳḄᐙᑖR⌕ᩩUV!⌲XYZᑨ<.19.BD[ᫀ]17⚓_58⚓

27ூᑖ᪆௃abA,B,e◤᪷h௃>abAᐘz&ঞ5w=%z}.11§5ww,z¨law®S=|2||a|-|^|sin^2a,b^,ᦑ¯u°w=z±°ឤᡂ²abB,2°=}ᓱ³´§/³Bz8wa=W•}sinz5,¸wᦑ¹°=6ঞ&ឤᡂ²abC,¼£=¾,¿b¨0,z5+^w0c=zl+2w|^|-|c|sinQb,c^,za®cw+^®cj=|A^|•}sinz5,0w+W•}sinz5,w=z1+¯w-|c|sinzÁ.,ᯠÂ+5wÃ1=z¸Ä+z5ÅswHឤᡂ²ab஺]_xw+x%Ç᝞É}PÊ2+42/ஹ2=Ë+Ì-0ᧅ=gᱏ+ÎÏÐ+ᡀwXzÒ஽+x%y=£+ὃு²-2%஽×|R=ᡭRXØÙᑣa®b=|x,%—஽X³ឤᡂ.ᦑ⌱BD.[ᫀ]18⚓_58⚓

28ூ*௃,⚪ὃ/ᔣ3ḄÝ56Þᑣß6K>⚪ḄᐵJAbGE⚪.20.ACDூᑖ᪆௃UUUUUUUUIU᪷h⚪àᩩUᓄáᑮ4Ao=+2AB+3ACpᑨ௃UU8LUUUUUU1᝞ëᡠí஺îïð஺ᡠᙠ0☢ᑁX¿A0+206+30C=0uuuuiuuuUUIUUULULIi—~páAO+2O6—2QA+3OC—3QA+50A=0ᓽAO=2z°B—QAw+3z࠷—OAwuuuiuun3uunᓽ4!=2!+3ãôᡠõAO=—A8+öAC,ᡠõAKḄ²24ᙠïð஺Êà஺î8CḄÊUUUuuuuuuiUUUUUIUUUUUUU1ᵫAO+2OB+3OC=0>pázAO+OC+2
08+OC=0ᡠõយ=-2
!+Áw=—4!ᡠõAOøBClḄÊᡠõB;ᵫ!=-4জpá|ú=4û|¿AC7/OD,ᡠõ——=?—=—,ᡠõOE=—EC,pá£C=—BC,ᡠõXECAC445EC2q-ADxBEsinZAEBᡠõ=V--------------------------ᡠõC²'iAOC±ADxECsinZOECEC22uuuuuuuuuruutuuuuuuᵫA0+20B+30c=0,páQA=2O6+3OCIuun,.uun,»´ýî|஺p=|஺஺|=1,¿03_|_0(?uur2uunuun2ULBHuinuuuuun,páOA=2OB+3OC=403+12O8OC+90c=13,ᡠõᡠõDKḄ.ᦑ⌱ACD.[ᫀ]19⚓_58⚓

29D\EBCூ*௃,⚪þ⌕ὃ/ÿ☢ᔣḄᭆᔣḄឋᔣḄᦪᔣḄḄᵨᐸᔣḄឋᑣ☢ᔣḄᦪḄ!Ḅᐵ#Ḽ%ὃ'(ᳮ*+,.21.ABூᑖ᪆௃ᓄ/0ᑮGj>=
%+1—2«„-32.*342%+32•6᪷6ᐳ0ᑮ4832஺“33=0,ᓽa,ᵨ+3=2
஺,+32,a=2n+l-3,>?ᑨABC⌱⚗0ᑮᫀ.nூH௃JK.L—2424+32-ᵨ+Ẇᦑ᳛=4஺,ᵨ32%—
2஺“+32•Tlmᐳo,ᦑU+V2%-3=0,ᓽ4+1+3=2
q+32,%=1,ᦑ/+3=4x2"ஹᦑ஺”=2ᔣ33.4=24—3=13,AZ[;ᦪᑡ^%+3!`aᦪᑡBZ[b““=2e-3,C┯dbs“=4L^—3஻=2"+233஻34,ᦑ஺┯d."1-2ᦑ⌱fAB.ூgh௃⚪ὃ'jᔣᦪᑡḄk⚗ᦪᑡlmᙠὃ'opḄ+,qᓄ+,rᔠᵨ+,.22.BDூᑖ᪆௃ᫀt20⚓v58⚓

30᪷6⚪mwxyzᔠM⊤}Ḅ~ᦪḄmgP4x”X2,ᙠᙠ3CgP',0ᔠ~ᦪᓽᑨA.ூH௃ᵫ⚪mwxyzᔠA7⊤}Ḅ~ᦪḄmgP4᳝,y2,ᙠᙠ3CgP',0OPVOP1-ᙠy=V+iḄpgᙶ᪗40,12ᙠxḄgp,ᡠ“I!“ᚖgz”zᔠbxy=Ḅ¡ᙶ᪗¢£¤g¥¦m§qᙳ*~ᦪᨵªgᡠᙠ“2Ḅmg஺4¬2ᙠw2ᙠ3Cp,0®j.°ᡠw2!“ᚖgz”zᔠbᙠு=²ḄPgᙶ᪗40,12ᙠxḄgPᡠw3!“ᚖgz”zᔠbx´=411¶+1Ḅ¡ᙶ᪗¢£¤g¥¦m§qᙳ*~ᦪᨵªgᡠᡠ!“ᚖgz”zᔠᦑ⌱fBD.ூgh௃⚪·⌕ὃ'¹⚪ḄºᎷḄᑨAx¼½¾Ḅᳮ*ᵨmᙠὃ'opḄᦪo¿+,ᦪoÀ+,ÁyÂÃ⚪.23.ABDூᑖ᪆௃ᵫ☢ᔣḄᦪ0fAC-AB^\AD\\AB\,BA.BC=\BA\\BD\,ÆᔠÇÈÉÈÊḄË̽ᳮ0⌱⚗A,BZ[ᵫ|®Ïយ•ḄÑÒ0⌱⚗C┯dᵫÉÈÊᐰ`0⌱⚗DZ[rᔠ0.ூH௃ᫀt21⚓v58⚓

31fᵫ®•=|យÖ×0$4=|4஺|ØÙᵫË̽ᳮ0AC~=ACAB,ᓽ⌱⚗AZ[ᵫ.ᐗ=|||Ûkos8=|ÜÝÞᵫË̽ᳮ0BC=p.q,ᓽ⌱⚗BZ[ᵫÛ.=ß||àcosQ—NAC0)0,ᓽ⌱⚗C┯dᵫwRtMCDÉRt^ABC,ᡠ|AC|ÝC|=\AB\\CD\,32(⊟.ä)x(ᮞ)ᵫ⌱⚗A,B0CD---------2-------ᓽ⌱⚗DZ[ABᦑ⌱ABD.ூgh௃⚪ὃ'j☢ᔣḄᦪஹÇÈÉÈÊḄË̽ᳮÉÈÊᐰ`Á᫏⚪.24.é2ூᑖ᪆௃ê£==m®+dì==஻k+®0AA',B,B'ᐳá|Z—í=|ᨴH|=g,|à|=gᨬJ|ä|ᨬJ®ðA'ஹ8'ᐵyy¢xñᔠòwᓽlóḄᨬJô.ூH௃ᵫ⚪mAB^OB-OA':•ra=OAr=mAB+OA=(1-m)0A+mOB,b=OB'-nAB+OB=(1+n)OB-nOA,ᦑᨵAA,õ&ᐳᫀt22⚓v58⚓

32•••×ߟØ=|øó=;,ᦑùúIûᵬ|=;ᨬJI®IᨬJ,•••ᨵA'ஹ8'ᐵyy¢xñ||ᨬJð஺ᑮABḄýþG•ÿ=,1124ᦑᫀs.2ூ௃ᐵᵨᔣḄឋᐵᐳឋQ=OA=+,&|2-'=|)1=>!■ᑖ᪆-4ஹB'ᐵ==஻+"ஹ0A."Ḅ%ᐳ,2./0123||ᨬ567ᨬ58ᓽ.25.40ᫀ;23⚓=58⚓

33ூᑖ᪆௃ᑭᵨ?Ḅ@A,ᎷC£=(F*B=(—F஻)3=(஻,L,2=(/,0),1=(0,1),QRᩩT-U^ߟߟXUY1’ᯠ\]^_`⊤b^F+24+F-@,ᓽ-cd.e+44ூgh௃C"=(j,஻==(r,0),c=(0,1)ᵫ⚪o=p=k|=1,cd=0᝞sᡠu“.(Uw+஻2),xy+஻2=]mtᡠu஻=1U(U.+{=|mttᑣᶭ+-12=1=>/ᑐ2e+4'஻+4ᑣs=(஻,v)=x(m,/i)+y(-m,«)=(m(x-y),n(x+y))u=m(x-y\v2w2“4ᡠu——m=4=4ᫀ;24⚓=58⚓

34ᓽ164/2Iᐸc2=a'—1--^——4-Ur-U-Z+4r+4o+2"▣U7|—+—ᓽ+2/U=||=|+|ᡠu+2c|+U4=2a+\FS\+|S£)|2¡¢,S,££¤ᐳ3ᨵ¦+2c|+—2a+1/^,Z)|=,J++4>4¨7^¡&©¡=±23«¬ᑣ|(+24+g-4Ḅᨬ58¯40ᦑᫀ4-^2ூ௃°⚪ὃ²ᔣḄ³ᔠᵨᐵᙠ.¶☢¸¹ᙶ᪗Ḅ?¼u-ᑮᔣIḄᙶ᪗¾¿_`Ḅ@ÀὃÁᑖ᪆ÂÃuÄÅÆÇÂÃÈÉ⚪.,3315ஹ26.(——,—)1616ூᑖ᪆௃ᫀ;25⚓=58⚓

35✌ᐜ7-₹•ÏÐXÑ6ᵨCOSA⊤b^a4ᵫÕᓄ×sin2A•Øs+A-g/,cᔠYÙÚᦪḄឋᑡ¬Ýh¬Ý7-fḄ«8à.ூgh௃ᐜ7á3•âã•ã᝞sᡠbC஺¯åABḄᵫ.஺¯¤¹æABC᜜èᙊḄᙊÑᦑODJ.AB,ᡠuÐ஺.=|,₹05Øì,Ïᓺ•Jîïð=18,pᳮ-|p•|.cosØយ/=|õṹ÷ø2=32.AOAC^UUU1UUUUUU1ᵫ.AO=aAB+0ACீaஹBwR)3-4cosA_______________„2___________a=AO-AB=aAB+/3ABAC=184/+3disA=6sin2AᦑV_______,ᓽ“h-AO-AC=/3AC~+aABAC=326a4w4-3cosAB=8sin2A2ஹ11923A.üÝýᐭsinA.Øfa+Uÿᓄcos?A--r+-IcosA,ᵫ—IvcosAcl,2x2x\,J1,(2⚪cos~A—|—t+cosAᨵᨬ$%,&ᔠ()*ᦪḄឋ./0121323—/+—1(?3ஹ-[5,+3]-1<------8•<1:;—cos~A-7+=>0§4ᨵᨬ$%.ᵫB1<----------<1C2x128F2x1223315-----

36AூUV௃X$⚪Y⌕ὃ\]☢ᔣ`Ḅᦪ`aḄbc;ὃ\ᙊḄefឋ.;ὃ\ghḄij;ὃ\()*ᦪᙠlmnopᨵᨬ$%q⚪ḄrCsᶍ,ὃ\ᓄuvwᓄḄᦪxijgy,zᔠឋ{|,}~⚪.31127-L?2jூᑖ᪆௃ᔣ`Z)bc.Z+B+2ᓫᙊ;ᑮ£=—2,wᓄLᦪᐵ;ᑖ.ூC௃᝞;ᓫᙊ=£,OB^bOC^cOP=a+bOT=a+b+c^᪷ᔣ`yḄ]yᑣM5/஺48¡=஺47P//CO7P=C0ITI.ri.riifrr.τuiinIτminτlUiriτmr.iUir|τuiiuτQq==a+b+c=1,/.\OT\=\OB\=L\BP\=OA=1,m=\OC\=1,JᫀQ27⚓;S58⚓

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38ᑭᵨᔣ`ᜳàð/cosa=cosñ,ò;/tanc,ᵫ-e$4;õ3;;/tan/?,ᐭtana-tan¡᦮ᳮ/JᫀூC௃CM¬UaLᙶ᪗ÕU;ᑖÖ¬Lx;y;zÛÜÝÞoßàᙶ᪗஺2Bøùᡠ¬Nõ0,—l,2,Gõl,0,0,Eõ0,l,2,â/7õ,”,2õ0<᪷<1,B1<஻<0,/+஻2=1;AW=õm,n+l,0â]☢N”GḄyᔣ`ü=õx,y,z,ᩔ=õB1,B1,2,ü=õᑡ஻+1,0m-GN=0[-x-y+2z=0Bᵫÿ_______
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