2ข6-8-2-64

เสน้ ขนาน

เสน้ ขนาน เสน้ ขนาน

Y

6. เสน้ ขนาน

เส้นตรงสองเส้นจะขนานกัน
กต็ อ่ เม่อื

ความชันของเสน้ ตรง งั้ สองเ า่ กนั

(4,2)

2 P(4,5) Q(1,2)
4

1 (1,3)

(4,2)1 C11(CC2(,x((8212)11,,,8y8C)1))1(2=CD2C,8((((1D)D2−2,3,(2,8(8−)−,))422)D,,4P4()()(−DPxD4P2P2,(((,5(,4−44−)y,)2,5225PP,)),)P44Q((=))44QP,,(P55P=C4((Q1))Q4,54Q,2,(2−(−51)1)21,),=2Q2QD=QQ5)5)41−((−==11=21Q,,5452242=−(−−))−8−112121,−(2−==4)211) = 1
C(2((,x821),,18yC)C1 )((2=---2D242,,8(8((D(1)mm−),(32))−),P==24,)DPD4PyyxxP)PP(CC1111(((−PQ−−−−−x4PCPC2C(22,Qxx,45yyPP,Q,422422,)y5)QQDD)2==)PP)PQPCCDQPCDD===QQ1133==(4Q55−−=44−−22=(C=,==−−4Q5(−−88−−5Q44412Q22542,)211−−Q2((−DD,(−28Q−−−D−28−18==−44==21)−(,2122)−(−−112(=−))=4=−−−=D=4=2)4==5215Q2422)1511433)211)−=−88−−−==(−8−=1211−−C((211−(2−−,1−=442=48−22C2)1C))1−()−==4=D2111)
1 1 =1 C D
CD CD D
C
D D

C

P P QQ CC DD
CD
PQ

-6
-4

2 2 A(−2,−4) B(3,3)A(−2,−4)

Cmm(CA1D2B2,−==CCmmmm2−−−(C(1CA)1AD1D2B24B−,,−−=−−=−=2=625332−C−−−−−221mm1))==224D224−(CC−CA1mD−mm−−−−B−−−m−−66,((C(−5C=33CA=755753361AD1mmDBB2,=,=,−−−===(=DC−=D51=A−=)1=D2B24−−−−−)7−52−7−,5−2−(−−(−−−−−−−==175675)167575)62A425233242−−−,−−=,=2==−51A−−−(5A)=−−=66D242−−)7557C533)7557533−−−m−−m−−(26==A==75(7D5A6C5D33A,1DB−,−2−−(−==−−(−,−==5AB(BD(−−==−475756C75)7755765−−m−−m22)2,(−−−=,===5(AA175C755,A6)A,12D42−B−−)7575()75,75,==B−5A−−−−4==64AA)7575533)22)−−−B(1,(=)=BCBD−−242−(m−m−−3−−4−2−−(2BC6(A1,D)7575,56B33,3B,−B−,−===)===5AD4((423−)7537−−5−−−−))1(),,2B72457536−3A−−−),()=6=(5A3B533−)7575,BC3B=C=2D)A(,(−−−−−3(3(B75756,C−4,33,==)25DA)D),)7575−CC4D)BCB(3DD,3BD)(3C

AB CD

2 (1,3) (4,2)

(4,2)

2

3 4 (1,3) A(0, −1) B(3, 2)

3 3 C3(2, −1) D(4, 5A)(0, −1) A(0A,B−((013,),−21)) B(3B, 2()3, 2)
(4,2)3

3
CC3(3(2C2,C,C((−2(−((x1,21x2)1,−),1−,1−,)1y13y))113)C)CC(=-22=2--42(D(,D22((−,((C,(1(mD141−−4m,(D),,D13,(1253))45)(,))()4,−A))4A=5,=1,5))5A)A)yxAyxCAC11A11(D(−−CxAD−−D(xCAC24BA(B2(xy,,4x4y,A52D,A2B,y22)y5B5((B(2D)0)B=D42A0AB=A),A,A,)−D5−(B=13(==DC1=0)1==3BAD1A0B),C)−C,A−((A(−−==C((−A(−−−4=−=−201==24014112(()114,2C)−)−)A−−((,−)−2)BB20((A2−−B11−−20=43)2043A−=)−)11()1125D−)−)025B−−−)−)B(43B−,DB−−D0−B=4−B−3=5D2−5=43=(,B5(2153==31−25131331,=B=3)==,DB==1B2=((===2==()1−−()3((320)((13311((−1−,−−3==20,20−−−)−)21120112)11−)−−)−−)−)(()43−)−)−−−−−−20254343−−1143B2525−)−)C==C25B(−−=433===1(C3==,253C1213C3,13)2==)13DD C 10
DC C
D D

C 10

D DD
D
-6 A B C
D

-4

24

(1,3)
4 A(1,2), B 6, 7 C( 3, 4)

4 42 A(1, 2), BA6(A,17B, 2), 2 6,C677((43,12, 4) ) C( 3, 4)
B1 5
2 AxA267(AB1,C,yA22)(1,)A21B,12=(21),6,(72,B443)7,,B261,)76,
AB 1( 1 C( C3(,C4(3), 43), 4)
( x1, y1)4= (14,34) 72
21y14)A---4C426B=--24((((m1m3AA,,43)CB4A))A)=CA=CAA4BABBAyx1B2yxB1111A(AAA−−AAAA−−xAAABCB(AA43BAABCBC1ABBA2AAxBCy(CB,AxBy1B,CC2A22CA,22yAA)C2BAAAA12,ACB2=12)B1122ACB=B(2112BA,AA2)11AC21BC,BCA13AABA=A21343667CB(C)BA−44331−,21674312261,−672−1C(,7B,2442712241)2,12,1432431666772B=1,1=21)74367221121436−67−,1157AC132113C21C(11(2121CA3,,(34,BCA)43C(A),AA4CCCAA)3ACCA,,4ACAACA),AB,,AACC,B,ABB,BBAA,B,
,B 6, 7 C C
2 7
1 6 ( x1, BC C
BCCC
2 4 1 C
1 3
C

A, B C

(1,3) (1,3)

12 2 k (4,2) (4,2) B(3, 3)

A(−4, −2)

11 k 1k 1 C(−2, kk)k D(A5(,−6)4A, −(−2)4, −2A)A((−−B44,(,−3−2, )23B)) (3, 3)B(B3,(33), 3)

C(−C2(,−k2), k) C(CD−(2(−5,2,Dk,6)k())5, 6) DD((55,,66)) 10

11( x1,11y1CCC)C(((=−−---(462−−222,,(2(,(1kk,kmkx,))kk3)1k)),((=−−y421)))((ymx((−−((((−−11((=---−−−−−−(−−−D426A42−−DDD42−−7533kBx44522))(42m))((((m(5))))))2mm155−−m=====m−−,−−xy−−−−−−−,,,A−−−A6−−−−−−−−−AAA6637533A2k7B533521kB5)km(yk)775573333533kkkBBBB55))5−)−k2==========−C=−2((===============7)D((((−−)−1km6(1k6km−(−−−k1km(11k=kkmm−((4213kk−−−kyx426−−4−2−−Ck−−))k2−−−mkk−C−7)C)−112−−2Dm−CCC5−−))−227)7−m(−−6DD776)−)−−DDD−−−66))−−−6−−−−4−−6A−66−x66A6−−−−4753−3Ak2B755363−−kB56,2A6657533kB5A2x5y,AA5=====55=====(=(2)−2=====y((−11km(km(k4k−−24−−−1,km=(−kk−k44−,C−−)2C5−−−27−,,7Dk−D)−−13C−2)6−26=6−−13672D−))−−622)666−−5())−564542,2=)BB(−(BB353,((,13333),,)33))

2 l1 ( 3, 5) (k, 4) l2
k k kkkkk2
l1 222 (((((366666,,,,,144444llll111l)))1)k)2l1(=l2l12)==34==l1−−−21=34=46(−5−3−21533−213−3411161(−51,−3(53llllk−3l222−k4k2lll3kl21111l((541,−===l21lll(12=(22l)1111=k)l1−2k4))1k======333444=)====5434==l21=34=)1)−−−−−−−−−−−222111−−21−44466621(((46−−−(555−5433361113555333−553(3333−−−21533−k333111331−,,,,−−−3,(((−1((kkk−−−−kkkk(−444kkkk4kk−5554444kl4,54lllk222111l212))))1114))))k1)l211)4) (((kkk,,, 444))) l2 lll(4, 3) (((
2 2(k, 4) l2
(6, 4) l 22
lll 222 2
2 l1 l2 l2
l2
(6, 4)
(4, 3)

l2

l2

lll 22
2

l2

2 l1 1 k2 (C(−l)2kkl2kk1,=l1kkkk(==34)6=kk(,−6−4−21,1111)464(−5311)5313−131,−(((k−k4−−kD54l2142)1)(3333))m5333−−,1111−−322−22A61122ll75l33l1kBkk5)kk22ll2122kkll2ll22l12ll1k2(12(12(l(k=====2k=(k=11k2l2k=((=l223kk2232==31l3kl1222(33lk1)l=12)ll)()1,2=llkm(k1))3===k314==l=1=314==234==34113==3344l1=)−−l=)1k=34−−−=C−1−=−34−−2−−=4−l721−2D1221222−111−))−4−66−646446−64261−−2513−24133653k31221612155332kk134k5211163k153k11−3353−2153k33A1335211533k11313111−3−−3(13(1kkkl−kkk4−−44k(kkkk2k44kk4k4k−kk4kl2241121kkkk2121121)k,4k21l21−1) 2) l 2l 2

(6, 4) 1 (2) 1 l2 B(3, 3) (4, 3)

lll22lll2l22222
lll22lll2ll22222

3 A( 2, 1), B(1, 0), C(4, 3) D(1, 2)

3 A( 2, 1), B(1, 0)Y, C(4, 3) D(1, 2)

3 A( 2, 1), YB(1, 0), C(4, 3) D(1, 2)

Y

X
X

X

3 A(4 42, 1), BY(1(,2(02,3),3,))C(4, 3) D(1, 2

3( )

2
2

-10 -10 AAAAAAAAA((A((AAAA((((((AAAABBABA22B22BBBB,,B,B22222,==2==,,,,,,======CC11C1C1)))CCCC1111DDC)1DC1D,,,CACBAABA)))),CABA))DDDDACBADD,,,,,BB,BCAABACABAACCBAABCCDDBBDDAABCCBDBDABCACDBDBBBBB((CCCCBDDDBDDBDDBDB(CBDDCDDB(1111(((((,,(,1111,11000,,,,0,,BB))B)00000B)-05,,,BBBB))))CC,B)CB)11C110000133144003,,4,,110CC0,34C-,CCCCCC5C111111110000000013333144404013==14003CCCC=11422C((12=C(12(44====4=11112222=1((((24(12AA2244(112A22331141,,231A2,4444412314,4AA3322A2A24444DD1111A232223332311141D1,,,,232A143,122311D2,121)))3333DDDD3)D3D222211112121))))))11113311133131131313111111113333111113331311313113131111111DDDDDDDDD((D((1111(((((,,(,1111,11222,,,,2,,)))22222)2))))())x(1x---462,1---426,yy11))==((((2m2m,3,)3))=)= X

11 55

23 ((xx22,, y2 )
yy11 −− yy22 =
3 xx11 −− xx22
4
1

21

13
12 1
21

= AD

(4, 3) D(1, 2)

EE DDDD AADEDEBB EEEE AACC DEAAAABBBB AABAACACCCCBC DDEE ABCBCDD DE EE ABCBC55,,าDEE626121รE))(ะB(,/B(,5(5/52CB5DะDDCB2B(EEC2(7(า6BED1ง,2)7(mm(,2-B612112)DB,)(D,22E,2CEY-61)216E(0()B(,4)/425(4)20,54Y/2)CD6C2242B032)(1C)20277(2A6BAD,263A,0)C(B2mm(0B(D2-)76(61D2D15),(),BD,,CE362742EA,ECEYA))(A)),76(EA0(A,104(0(A(5C7))25)05(14142,CD5,,,621,2246C21621643)2))())(B((1)(),B(B(,7X,52E555A5,5CB60C3CBBA(0B2)(22(76((25,2,7(,2277C3((6B4BB1))1,1()D(,,,7((,,E-6A1าา2,-1E-161612201X1รร)า,)C7))),),)2ะ14ะา2ร142,22Y2,,)YรYะ))6Dะ6ะ6DDDD000DาะาDEEE)2A0ระ)2DD0)รา2า0EDะXDEBEDDาาาาะงงDาEC2รรรรEDCC2าCง2DะEDะะะะDงะา)BDD()DD)(า(EEร221E7CD7EรD7ะะะะาDDDDDEDDEEา2B1B,D((DEะ,B((EงDE,2งE//EA0EEE55((A(0(55CAB(ะCE0าาา(//DDา((22BC(()6ะ5(DEDE(ED(DDE)งง6ง5(2ง/)/62(D2255(CE2BBED,5521CE/,5,า2E2,,,2,,CB3EDEา6CCC,3ง4E622C(77432DD6D((BEDB((D4EEEง)(E)626/2BB1C())/(5)2)7))5(5((/7C,,5EEE)))7E2/BCE((mmmm7E((((72D(B(((,E22--E(6(,B6611,///(20D,(/1mmB1555((02E5B55(5,20///52)))-E(mm61/BDBD)22,,C)21(C222E-()22612C(7,,)CC/((12EEY2Y22DB,71BBB(5(2))1BB52)/2/6BDB,665,(((2CEY5,2)/CCC002(,mm,2222CC)44777EY(mm(((6227)1B2-(6166BBB2-0))6221162006(BB10),,,C))DD7BD,((4()mmmmmm,2)2)DB707,(mm(2C222---2X666,111222C)6XB27E0Y2)D-(4242,11161CE)Y)X16)B33)2))D,226)7BDBDBD,,2,(mm))2)624(BD,0,22222)221-120())mm6,,,12CCC242EEEY2YY)41),)))C2-EY22613)220)2)11666))((2ADBA,06(000)D21)662233BD,44D,AA0CE(YA24((2020B)B2(,((C)))222426E0Y0032A6(24A)7766(()20222(260B0D3DD36)((2A554),,D((00B76(1,,(2224,,1))24224420CC257363(6622,4244233)22,2220DA5,)))),)3A)C)2((3,66D3)))11104)),A623C77243A6(100B)E)E4)22(((A2A0B)3)24,,(11AAA76)2())7(00)276(32)E26A662333A510))AAA7CC,,7752))E6,(3,((0B00BB1A01414),A2,(((141,41)C,3(06BA76776C6C(((0432(6)22242146376A5(5C514)7)A,)(,2,))),14,,(6(0B3514,,,)A(7,CCC333E,7666(0B76(A44,4E(2,AC36,1765)())04)))(((,20))),5C),)777),(CC76E3E)E,146X)XAAA144,147,,,11C1463E6)0040A8)(,X1)CCC7)7)087)())X6)E14141414A1414C77,)1E14A0814,1C70)14C174)X14X1422XXX21X17XX717

DE D E AB AC ((( ) )) DE ABCBCA231B4ED=)C()2(7(21526B22,,)7(mmาBBEE2+-6=6121−)(ร1C2)C)BD−,ะ(22(B(,6C,EY1)1556(ะDD)=)CB0B(4−(22E2−)202C(าD7ED+4ง22=742()(3)B2)1(−2า10),2D(mm,E42Dร)-161A+21E2+163ะB(A4)−(E(BD0,B()−/(652(276((5,C2/C(EY25()ะ3,2DD1,7B6,+2C1B360)4))E=−))(4CC)22า12)27027(621E)าEDA6Bง,D1008,ร+)=7==(mmCC7B24)24)=214-61ะ14)(12)2C2)((DDB−,E12D272,CEะDD)1E1B=1A2A(,6(((6(−3E0(−AD6/074E−B(5((X0BC25า(D/)202)27164(DE02งDD+75E,7−D2(A1),B)2,42C)3−61342E22)CA62)2)(D7(10)(8)E21D6+B7ED2=B5)E)=A+A)(E,,2)76(23mm,(0BA5(2ACE(((0B2-C/61())65(2(111,1476(51+42/CC+E2)22E5DB,6),211),2(=,2CB,32C621EB),(47−)−((B)()(6,4()2C5780C26B5E7=4(A021(,1C)B6B−)08)2X(02)C7(2,2))72(1mm4−B14(232-61)21142+1=27)(C2)B3DB,2=B)162−2212,10)C(E(,,)(m,22-)6(-+X621A712=016264)31)(,)D),)((01B4)222(02722D,−276EYY2)),+=68E=,2+(240C334)−2าา)4)20)((2010))รร4D7262D1ะ)Aะ,A61+=C2)42028C637−4C227()0B)((14ะ(ะDD14D76B(B7220+BA+7,AEEA,6,))CCาาCB6C33,A(4048ED22B(0Bง2ง1()A(0C))6(27)6=5===)=AX,B(,1,2,085CD1D36E4CC7DD),214)C7)14(6=EE)(7((E)6EE6((A((//,D5((−D05−71811D/4/EA1212C,)1121147778E110772E1CB1B+77=E))−6XD==CC2D2=71=4(D21212==7+6B6E7+12,E)7))7(mmm1=2BBE11X122-2216121)()1121721C2)C2221BD,=B772+B2X272,)CCEB)6B12(621(C(80127214C41(1(21C=7B)20227−7=BBC2D2402C3322)6C)E10)10)2422)1)A2263+7=(0B0(76742,,C3(=4A−)()B766E8A,11C088)6C77)214B14148+C=(2222X)112

แ ฝกึ ดั คร้งั ่ 2.1

เสน้ ตั้งฉาก

เส้นต้งั ฉาก เสน้ ตงั้ ฉาก

Y
−1

7. เส้นต้งั ฉาก

เสน้ ตรงสองเส้นจะตัง้ ฉากกัน
ก็ต่อเม่อื

ผลคู ของความชนั ของเส้นตรง
ั้งสองเ ่ากั -1

12)11 1 11((−(11,,32222(()),355)32,, 66))(1=(((1,2,−(2)5510),,,66−6))(()1−((23(−1,122,2,(2)2,322())3,32(115xxxyyyy3)66,5)311111122344362233,32(2=(16)(()(),((1=−21yyxyyxx6626(,=()551,,22222222−(05())3)−0−−1)((,515)1,0(6,,1,2326,,326)62)(6)()()21)(()1(,1,(,32((,2232((3222))((,21)((,5)(31,022(1,3xxyy,(23253),,,,6(11256)xx=yy1111(32=,3))3246332,62,3,)(21111)6232,243)))3623−()2−)3)5))()(15),1yyxx(660(,6(5yyxx222266,60)5)((222265(()(10(51)0xxyy(()3,50,6,(0,6((1111,2,233246622,30026)(2(6)5),6),,,(0)()5),3)6603,,6yy5xx)((16,))6)6xxy623),5(662222))1112360423)23)22)(1xxyy(,2)3(12xxyy23(16(xxy323,1111623yxx324111162623363423)111222335342222223(1()2xx(yy)5(03)()(106xxyy(0)2,311112(3yyxx4236,6623yyxx6,111126236yxx234562262223665)22222()05)2)2202))()0)yyxx,66(yyxx536622220(50)22)22,00)6,(1xxyy(1236x)y23(66(11112)3112243322342322-2()1()2(,(,(yyxx6yx360663522225),222)0)0)62,,)33)) ( 2, 3)
(0, 6)
2) (0, 6()0, 6)

(0, 6)

-1
-1
-1
-1

ว าาาาว 33า3−3(()616,,((5(5)(()((1133352,(,,,,16132322,,,)))))5)−33())521,22513xx25y132xx5yy13xx55yy223(22((1111111)=6111111352511136525152,,,3(3653(5−65653−−))yxx((yyxx1)(yyxx2(23(231532223222222552222(2(,,(22,2613323(((23,,((11))))())11−511==,,(,,,)1−33,61335,,−−3,))))5235)311)))32513xxyy25313xx=yy)2(((()1111111111=1135526(((1((1352−1((1,,366113(6(5,5−633,1511,,,,61(3)3yyxx)1yyxx,,(655(,2,3((23))32222,522))22((((2,33(25)133332(333))),),))),))3113322,,,1,)))2((,2)66(()336,,6-)((,)553,12513xxyy((5(5)))1111))1135233(33,,32513xxy3625136xx5yy,,22,22111,221213525))yyxx1111(1(13552(523)))22223)((653265333yxx(yyxx23,(,)2331222222222,,2(22551133xxxx2yyyy))32225511332325xxx1yy3xxyy11111111))11112233552252)11111311111111133525552113255-,33)66551333665565yyxxyyxx((22332yxx5yx133yyxxxxyy((22222222(223323((2)2222522212111122x22y113552(33222211331313255))311(65))),,136yyxx,(23322yx22(23,33(222233))2)))33), 2) ( 3, 2
(6, 5)ว 3 ( 3,
1, 3า)3 (ว1, ( 3, 2
(6, 5)ว
-1
1,13,) 3)

1, 3)

1, 3)

 − 5  =

2 A(−4,3), B(−1,2) C(2,11)

2 22 AAB(−4AAAAAAA,3AABBBBA((((AAAAAAAA)B(CC−−−−AAAB,ABABB(−A(((AB4444BB−(B−−−(B4(AAAA−,,,,−4(3333444−A,BBBBBBBBAAAA(4−3A,−A4AA))))B,B,,A4CCCCCCC3(A3(33,ABB,,,)A1BBABBBB−A,AA−CC432A,)3,B3BB3A)BBBBBB))CA,C3BBCCACC1C)C,)BB−4A,2,,B−CC−C)CC)−,)CCB(((()BCCBB1,)−1,((((2(B−−−−−(1−−−−B1(−(B−2((((((((((−(11112−−−−−(−−−((44442222(−1−−(−(3333BB333−1−−(−,,,,((A−−(42(1111)))))(((−3B−3((1−−−−4444−2222(−1−−−−−,11CC−−−−−42−−−(−))))−−−−−14)2(B443(22−B3C(−341(B3233BB33)))),(−−C−−1111−,42(1,,)−1))))−−−−)(−4111121311B−))34242222213B3−−−−−−4)C−((((−244,22−31B311142−C)−−−−CC)−−−)1111,3−1)B))3−−−−−42(2)1)−−42222,(−1C))=1)−−−C4−=1112))−1=1)−2222−)))−C−−1−21−−42−2)()22−22C−1−1111)1−−(())−1−11111)−2−11112)A−AAA2−)−−−2(1))))−−12−1(22−2)−−−,11−21(1−122122−A−(211)1BBBB2−1CCCC1122−2A8619332)A21AAB11C))21AA1B)C)BABBC((((CC))A=======)B====CB(2222BC==B=C(B2(((=−−−−−−−−=−−−−=−−,,,,===========2(1111222−−=−C(−=−,====(888866661=11112999933333333=−−−=−34−,1111=3−−−−2=−−−−−,−−−−,,18621933111−−−−−−,−−−−))))−86−−19133−−,868816693−311991−3333−−−,111)18619338619331861933)BBBB8)611))933====1====)BCCCC=)=B)BBB=34434334C3333=====B==BC43=3BC=CC==B=43=−3=C433443C3=33C34C33431433343
2222222
22

−1

−−11− 1
−−1−1−1 1
− 1−−11
−− 11

AABB BBCC

3 (1, 3) (6, 5) ( 1, 3) ( 3,

ว า5 65)mlm55555121mmmmlm22mmmm122 ==1==1111(−−−−−l−(((((11−16−−−−1616)ml6m66661)ml))))2mmmmm1llllmmmm1lm1111mm12122221111mmm2mmm22mmmmmmm2=2=2==222222222222l16=======−===========2−−−l−111−−−−−−−−16−−−−−−−l16−l−−−−lll−−−112l1111111111111611611116666((111166661,13,)l13l16)2l16llll11116666222226(6,l(52l1) y61 y2 6 66666 l2

l1m1 l1 lllll11111 l13x2l11lllll22l2222llll1111165x2

m2 (− 3, 2) 5
2

5
y1 y2
x1 x2

3l2 2

1 ( 3)

m1m2 = −1 32
13
6) m2 = −1 5

−1

m1

77777(77−7((((−−−1m((24−−−(,)(m111mmmm222m−−16m22,),),)11mmmmm22−mm,)71m6667m2121,,))1222mm6−−−mmmmmm2,)66m1m)111222−mmm6−−1mmmm2((m11111222m−(mmmm=m2=m−1−12m1mmmk11+22mmm===m==m=mm11143m2+m211m1=1m1mm,2+4=m+++mk==51mmm==mm,)1112+m+,)++111=111m11+−4=4m4++12k62mmm+1)11−316mmm21122214++++125m111121−mmm−−−44−122−22mmmm+111=11−5−1−3−mm422−1411=1ml1m255m5=1==2−22m−−−−m=−2122=215−−224(44m−−11−===1111(2255=1====mk==−224−−==−−=−−=1=1m−===2−520==−(2(((446m=−=−−112=m=−==6===(4−−−==−5====1m=1−−−==+m=0−(=−(−(1((66612=−2−−=−−−1+−==1−−−==1=−(+(−−−611−−===−11−4mmm−+−−(((1111m1(2662)2m−)−=−−−−1−114m−12−−m111km−(−16−−2)m−−−121−11−12−11−1m−m)m226))=−−(−−−11−=1111m11−21152m−)−)−−))2−1m21−1112+1151mm2))(((2−)1=4==111−41=1=m)1=(k)),4mm=)m1=−=143m=,2=22=−)6(((==4m44=−2−=+21((=mm=−4=−+mmm,,,−)1−(6m44−−=m−+++−,11211−(+−46−2mm=−−+−m,,−11221−k−1=m43,−++1m11+++122122−)1+−−1−21+11m)11243−1+−1++−)m)))(221222+111(=−2)−121m2())=2224)m21((2m4,(((−−+11)m,−−−−(l,+111111+lk−421111,,1,+))11,2444)1)))1+2,,4,)−−2444−−−m24(mkkmm),mkkkk,k44451mm1m,,,4++11m,,,,,,+++111544551,11144,,511,,,,)22,5511222451(=−m==1m344===m,kk),kkkkk−−451kkk1−−−,2,++11,−(mm11++1115+++11114111−+++1,,,2kk22,111233444ml1kkk,11222122,,2433311m2111,6m2=111,==,,44)=m==,==4==122,===+66==6====26===−16,,===+11+,−4−−m00−−66m−−2−−−m6((000−−−−−−−1332431(((−−−3111kkk333,52−−−111−kkkl441==111===2=−−−+1111,=())−−−,4−((=)))−−06−(2622−(((2kk−−2223,kkkm12−116++1)+++1−)=(22=222mkmk2))−)))mmmm+1,m1++5m−,mm,,mk+2,535251151,,,=)=255==5==−−0−m−(33−1k,m51−),(52k m

+

แบบฝกึ ทักษะท่ี 2.2



4l 3 l
4

4 4ml1 44 l 3 3 l 33 m1 = 3 l l
4 4 ll
m1 4 ll ll
m2 mm12 l ml1 =44 3 3 l
4 mll 1l = 4mm11 33
mm1111m2 = −1 == 44
m2 ====4343mmm−−−−112221143mmmmm22222344343======mm−−−−−111−mmmm111mm114343222222222 ======43==43 l
l
m1m2

3  m2 −−11
4 −−11

m2 44

−−11 l33 −4
3
−− 44 l −4
33 −4
ll 3 3 44
−− 33

6 l1 l2 l1 2 l2
l2 5
m1
m2 l1

m1m2 = −1 5
2
2  m2 = −1
5

m2 = (−1) 5 

2

l2

8 A 4,2 , B 1, 5 C x, 3 , D 0, 6

x

AB = 2 − (− 5) = 7

4−1 3
CD = 6 − (− 3) = − 9
0−x x

AB CD

 7  − 9  = −1
 3  x 

− 63 = −1
3x

− 63 = −3x
x = 21

x = 21