UNIVERSIDAD NACIONAL DE INGENIERIA
CUARTA PRÁCTICA Califcada APLICACIÓNES VECTORIALES Alumnos: • • •
Romaní Morón, George Anthony Salvatierra Ayala, Manuel Gonzalo Murga !arga", Al#o $lí
Profesor:
20081261I 20081248 20082646A
ringa" %&'iga, (e"&" (o")
Sección:
S
Curso:
Geometría *e"+ritiva
Fecha de entrega:
10 #e noviem-re #el 200.
FACULT ACULTAD DE INGENIE INGENIERIA RIA GEOLOGICA MINERA MINE RA Y METALURGICA METALURGICA (FIGMM) UNI – 2009
APLICACIONES VECTORIALES DE LA GEOMETRÍA DESCRIPTIVA 1. Repree!"aci#! $ec"%rial Las magnit!"s #$si%as s" &"!"n %'asi%a* "nt" +tas m%,as !"n+mina%i+n"s* !" !+s man"as. "s%a'a"s / "%t+ia'"s1 Una magnit! "s%a'a "s 'a " s" """ a 'a magnit! "n s$ misma* "s !"%i* magnit! "!a!"a "n s$ misma " &"!" "sta "&"s"nta!a &+ n sim&'" n3m"+4 as$ s+n magnit!"s "s%a'a"s %an!+ ,a5'am+s "n a5sta%t+ !"' ti"m&+* t"m&"ata* +'m"n* masa* "t%1 Una magnit! "s%a'a* &"!" tam5i6n s" "&"s"nta!a g7%am"nt"* tant+ "n &'an+s* !iagamas* "t%1* "&"s"ntan!+ a 'as magnit!"s 'in"a'"s* 7"as* + +'3m"n"s1 C+m+ "m+s "n "st" %+n%"&t+ "s%a'a nn%a a im&'$%ita 'a i!"a !" !i"%%i8n1 Una magnit! "%t+ia' "s 'a " a!"m7s !" t"n" t+!as 'as %aa%t"$sti%as %+"s&+n!i"nt"s a' %+n%"&t+ !" magnit!* ti"n" a!"m7s "' %+n%"&t+ !" !i"%%i8n4 + s"a* i!"as !" !"s&'aami"nt+* "'+%i!a!* a%"'"a%i8n* m+m"nt+s* #"as* t+si+n"s* "t%1 Una magnit! "%t+ia' s" &"!" "&"s"nta g7%am"nt" &+ n s"gm"nt+ "%ti'$n"+* %+m+ na :"%,a + "%t+1 La '+ngit! !" "st" "%t+ !"5"7 s" &+&+%i+na' a 'a magnit! "n "'a%i8n %+n s ni!a! !" m"!i!a* "s !"%i* " "sta '+ngit! s" ";&"sa &+ na %anti!a! !"t"mina!a4 tam5i6n "s &"%is+ %+n+%" "' &nt+ !" a&'i%a%i8n !+n!"* g7%am"nt"* "s" "%t+ &a"%" &ati* + &nt+ !" +ig"n4 / asimism+* "' t"%" !at+* s"7 'a !i"%%i8n !" "s" "%t+* "n 'a " s" &"!"n %+nsi!"a '+s !+s s"nti!+s1 Est" s"nti!+ !" 'a !i"%%i8n !" n "%t+ s" in!i%a g7%am"nt" &+ 'a &nta !" 'a :"%,a1 Can!+ n "%t+ s" in!i%a !"s!" "' &nt+ !" a&'i%a%i8n* t"n"m+s /a na !i"%%i8n ma%a!a %+n n s"nti!+ " "s "' " in!i%a 'a '$n"a !" a%%i8n !" "sa magnit! "%t+ia'* / "nt+n%"s s" '" ''ama "%t+ '+%a'ia!+1 <"+ si "' "%t+ s" m+i"s" "n na !i"%%i8n %+n"ni"nt" " #"s" &aa'"'a a 'a " a' &in%i&i+ t"n$a* "nt+n%"s s" '" ''ama7 "%t+ 'i5"1 Las +&"a%i+n"s " s" ,agan %+n '+s "%t+"s s+n in!"&"n!i"nt"s !" s &+si%i8n* &+ "s+ '+s "%t+"s 'i5"s s" &"!"n "m&'"a %+n &+&8sit+s !" sma* "sta* "t%1 En a'gnas a&'i%a%i+n"s* sin "m5ag+* '+s "%t+"s '+%a'ia!+s s"7n '+s " t"ngan " a%ta s"g3n na '$n"a !" a%%i8n !"t"mina!a* &aa na s+'%i8n " s" 5s"1 Un sist"ma !" "%t+"s &+!7 s" %+n%"nt"* %an!+ t+!+s '+s "%t+"s %+in%i!an "n n &nt+ %+m3n !" a&'i%a%i8n4 &"!" s" &aa'"'+* %an!+ '+s "%t+"s s"an &aa'"'+s4 / &+!7 s" n+ %+n%"nt"* %an!+ '+s "%t+"s ni %+n%"n "n n &nt+ ni s+n &aa'"'+s1 E' sist"ma !" "%t+"s &+!7 tam5i6n s" =5a>+ +ta %+nsi!"a%i8n= %+&'ana* %an!+ '+s "%t+"s a%t3an "n "' mism+ &'an+ + "n &'an+s &aa'"'+s4 /
&+!7 s" n+-%+&'ana* %an!+ '+s "%t+"s a%t3"n* "n "' "s&a%i+* "n &'an+s " n+ s"an &aa'"'+s1 C+m+ '+s "%t+"s s+n s"n%i''+s s"gm"nt+s !" '$n"as "%tas* '+s &in%i&i+s !" 'a g"+m"t$a !"s%i&tia '"s s+n !i"%tam"nt" a&'i%a5'"s* &aa 'a s+'%i8n g7%a !" t+!+s '+s &+5'"mas " im&'i"n magnit!"s "%t+ia'"s1 &. S'(a ) re"a de $ec"%re c%pla!are D+s + m7s "%t+"s s" &"!"n sma g"+m6ti%am"nt"* &aa +5t"n" +t+ "%t+ " s"a "ia'"nt" "n "#"%t+ a' " &+!%"n '+s +t+s !+s4 a "st" "%t+ "ia'"nt" s" '" ''ama "s'tant"* / si '+s !+s "%t+"s !a!+s s+n %+&'ana"s* "' "%t+ "s'tant" "sta7 "n "' mism+ &'an+1 <"+ &aa "' sim&'" &+&8sit+ !" sma + "sta "%t+"s s" &"!"n %+nsi!"a %+m+ si #"an "%t+"s 'i5"s* &!i"n!+ s" tas'a!a!+s a &+si%i+n"s &aa'"'as1 L+s* s$m5+'+s / / / in!i%an* "s&"%tiam"nt"* 'a sma / "sta !" "%t+"s1 L+s "%t+"s %+&'ana"s s" &"!"n sma g7%am"nt" &+ %a'i"a 'as sigi"nt"s %+nst%%i+n"s* " s+n simi'a"s. LE* DEL PARALELOGRAMO1
En 'a ga ??-? s" !an 'as magnit!"s / !i"%%i+n"s !" !+s "%t+"s A / @4 s" "'ig" n &nt+ %a'i"a O* %+m+ +ig"n !" taa!+* taan!+ &+ 6' 'as &aa'"'as a '+s "%t+"s !a!+s* t+man!+ s+5" !i%,as !i"%%i+n"s* / "n "s%a'a a5itaia 'as magnit!"s !" %a!a n+ !" '+s !+s4 &+ '+s &nt+s ";t"m+s !" !i%,+s "%t+"s* + :"%,as !" '+s mism+s s" taan 'as &aa'"'as a %a!a n+ !" "''+s* &aa %+m&'"ta "' &aa'"'+gam+* %+m+ "m+s "n "sa ga4 'a !iag+na' !"' &aa'"'+gam+ #+ma!+ %+"s&+n!" a' "%t+ "s'tant"* "ia'"nt" a 'a sma g7%a !" '+s !+s "%t+"s !a!+s* m"!i!+ "n 'a misma "s%a'a " ,a5$am+s ant"s >a!+1 O5s6"s" " "' mism+ "s'ta!+ s" +5ti"n" taan!+ !"s!" "' +ig"n O '+s !+s "%t+"s* " si ,5i6am+s t+ma!+ 'as magnit!"s !" '+s "%t+"s "n s"nti!+ in"s+* !"s!" '+s &nt+s ";t"m+s !" '+s mism+s* + &ntas !" : "%,a* &"s si"m&" t"n!"m+s "' mism+ "s'ta!+ !" A / @1 LE* DEL
TRIÁNGULO 1
En 'a ga ??-2 t"n"m+s '+s mism+s "%t+"s A / @* !a!+s
ant"s* %+n 'a misma '+ngit! !" "s%a'a1 D"s!" "' +ig"n O s" taa "' "%t+ A* "n magnit!* !i"%%i8n* / s"nti!+ %+n"ni"nt"* &aa !"s!" "'
";t"m+ !" 'a :"%,a A taa "' +t+ "%t+ @* tam5i6n "n 'a !i"%%i8n* s"nti!+ / magnit! !a!as1 Si a,+a nim+s "' ";t"m+ !" 'a :"%,a !"' "%t+ @ %+n "' +ig"n O* s" ,a57 #+ma!+ n ti7ng'+ %/+ t"%" 'a!+* "&"s"nta A / @* %+n na magnit!* !i"%%i8n* / s"nti!+ ";a%tam"nt" iga' a' +5t"ni!+ ant"i+m"nt" %an!+ s" #+ma5a "' &aa'"'+gam+ !" #"as1 Si "n " !" ,a5" t+ma!+ '+s "%t+"s "n 'a #+ma in!i%a!a ant"s* s" ,5i"a t+ma!+ a &ati !" O "' "%t+ @ / '"g+ "' A* t"n!$am+s ";a%tam"nt" "' mism+ "s'ta!+4 '+ mism+ " im+s ana'$ti%am"nt" a,+a s" " g7%am"nt"* " "' +!"n !" '+s sman!+s n+ a't"a "' a'+ !" 'a sma1
% MÁS VECTORES1 La '"/ " ,"m+s ist+ &aa #+ma n &aa'"'+gam+ + n ti7ng'+ !" #"as s" &"!" ";t"n!" &aa sma t"s + '+s "%t+"s " s" &"%is"n1 En 'a ga ??-(a) s" !an '+s "%t+"s %+&'ana"s A* @ / C1 <+ n &nt+ %a'i"a !"' &'an+* O* &+ ">"m&'+* s" taan '+s "%t+"s A / @ " a' smas"* "m&'"an!+ "' &aa'"'+gam+ + "' ti7ng'+ s" t"n!7 "' "%t+ "s'tant" A / @* "' %a' s" sma7 a s " %+n "' C* &aa +5t"n" +t+ "%t+ "s'tant" t+ta' "n A / @ / C* " "&"s"nta7 'a sma t+ta' g7%a !" '+s t"s "%t+"s !a!+s* '+ mism+ "m&'"an!+ &aa'"'+gam+s " ti7ng'+s* si"n!+ &"#"i5'" "' ti7ng'+ &+ s s"n%i''"* /a " "n "a 'i!a! s" 'imita a i t+man!+ '+s "%t+"s !a!+s a &ati !"' +ig"n O* "m&"an!+ %+n "' "%t+ A* "n !i"%%i8n* magnit!* / s"nti!+ / !"' mism+ m+!+ sig" %+n '+s +t+s !+s "%t+"s @ / C1 A &ati !"' ";t"m+ !"' "%t+ A s" taa na &aa'"'a a' "%t+ @* / !"s!" "' ";t"m+ !" @ +ta &aa'"'a a' "%t+ C* &aa t"n" s+'am"nt" " ni "' &nt+ ";t"m+ !"' "%t+ C %+n "' +i g"n O* #+man!+ "' "%t+ "s'tant" 'a '$n"a " %i"a "' &+'$g+n+ " s" a #+man!+* ''am7n!+s" a "st" sist"ma "' m6t+!+ !"' &+'$g+n+1 Si"n!+ "st" m6t+!+ "' &"#"i5'"* s+5" t+!+ %an!+ ,a/ " sma m%,+s "%t+"s1 SUMA DE TRES
SUSTRACCIÓN DE VECTORES 1
S&+ngam+s "n 'a ga ??-B " ""m+s "sta '+s "%t+"s !a!+s A / @* si"n!+ "st" &+5'"ma simi'a a' ant"i+* /a " si %+nsi!"am+s* &+ ">"m&'+* a' "%t+ @ " t"nga "' mism+ a'+ / !i"%%i8n* &"+ s"nti!+ in"s+ "' &+5'"ma "!a "!%i!+ a sma '+s "%t+"s A / -@* %+m+ '+ "m+s "n !i%,a ga* 5i"n %+n "' &aa'"'+gam+ + %+n "' m6t+!+ !"' ti7ng'+1 Es !"%i* " si !"s!" n +ig"n O t+mam+s "' "%t+ A* "n s !i"%%i8n* magnit! / s"nti!+* / &+ "' ";t"m+ !"' "%t+ A s" taa "' "%t+ @* " ,a/ " "sta* "n s misma !i"%%i8n* magnit!* / s"nti!+ in"s+ a' " "sta5a taa!+ %+m+ !at+* 5asta7 ni "' +ig"n %+n "' ";t"m+ !" -@* &aa t"n" "' "%t+ " "&"s"nta 'a !i#""n%ia g7%a !" '+s !+s "%t+"s !a!+s1
3.
Descomposición vector coplanar
de
un
"m+s ist+ %8m+ !+s + m7s "%t+"s &+!$an %+m&+n"s" &aa #+ma n s+'+ "%t+ "s'tant"1 <+ "' &+%"!imi"nt+ in"s+ n "%t+ s"n%i''+ s" &"!" !"s%+m&+n" "n !+s + m7s "%t+"s %+m&+n"nt"s1 Si "st+s %+m&+n"nt"s s+n %+&'ana"s* ,a57 " %+n+%" 'a !i"%%i8n / magnit!1 <"+ "' %as+ m7s %+i"nt" "s "' " n "%t+ t"nga " !"s%+m&+n"s" "n !+s %+m&+n"nt"s " t"ngan na !"t"mina!a !i"%%i8n1 En 'a ga ??-(a) "' "%t+ !a!+ A ti"n" " !"s%+m&+n"s" "n !+s %+m&+n"nt"s " t"ngan 'as !i"%%i+n"s OM / ON* s+'i%it7n!+s" 'a magnit! !" %a!a n+ !" "''+s1 En 'a ga ??-(5) s" " %8m+ &+ "' +ig"n O !"' "%t+ A* s" ,an taa!+ 'as &aa'"'as a 'as !i"%%i+n"s !" '+s %+m&+n"nt"s !a!+s* &aa '"g+ &+ "' ";t"m+ !" !i%,+ "%t+ A taa 'as &aa'"'as a "stas !i"%%i+n"s* / %+m&'"ta as$ "' &aa'"'+gam+ !" #"as* %+n '+ %a' "' "%t+ A ,a "!a!+ !"s%+m&"st+ "n '+s !+s %+m&+n"nt"s* + sman!+s* A N / AM %+n '+s
s"nti!+s " s" ma%an &+ 'a :"%,a* "n %a!a n+ !" "st+s %+m&+n"nt"s1 En 'a ga ??-(%)* "m+s 'a !"s%+m&+si%i8n !" "s" "%t+ &+ "' m6t+!+ !"' ti7ng'+1 <+ "' +ig"n O s" ,a taa!+ 'a &aa'"'a A M a 'a !i"%%i8n " ,a !" t"n" "s" %+m&+n"nt"* OM4 &+ "' ";t"m+ !"' "%t+ A* &+ s :"%,a* s" taa 'a &aa'"'a a' +t+ %+m&+n"nt" ON* "n%+nt7n!+s" '+s !+s %+m&+n"nt"s A N / A M "n n &nt+ " !"t"mina "' ti7ng'+ 5s%a!+1 La !i"%%i8n !" %a!a %+m&+n"nt" s" !"t"mina tam5i6n "n (%)1
+. S'(a de $ec"%re i"'ad% e! el epaci% L+s "%t+"s " "st7n sita!+s "n "' "s&a%i+* s" &"!"n "&"s"nta "n 'as &+/"%%i+n"s m3'ti&'"s " %+n+%"m+s1 Y ss '+ngit!"s* !i"%%i+n"s / "'a%i+n"s "nt" "''+s* s" &"!"n !"t"mina &+ '+s &+%"!imi"nt+s / m6t+!+s %+i"nt"s !" g"+m"t$a !"s%i&tia1 ESTA,LECIMIENTO DE LA LONGITUD DE LOS VECTORES1
Fas" . Esta5'"%" 'a '+ngit! "!a!"a !"' "%t+ %+m&'"tan!+ 'as &+/"%%i+n"s1 <+ 'a '$n"a O T A* &+'+nga!a si "s n"%"sai+* s" taa a "s%a'a 'a '+ngit! "!a!"a !" 1 ni!a!"s !"' "%t+ !a!+* &aa "sta5'"%" as$ "' &nt+ aA1 L+s &nt+s a T / aF &"!"n a,+a s" '+%a'ia!+s &+ "' sist"ma !" a'in"a%i+n"s1
SUMA DE ESPACIO. La
DOS VECTORES EN EL
ga ??-?0 n+s in!i%a "' m6t+!+ &aa sma !+s "%t+"s "n "' "s&a%i+1 E' !iagama !" #"as* !" 'a &at" ii"!a* in!i%a 'a sita%i8n / !i"%%i8n !" '+s "%t+"s !a!+s A / @* &"+ ss "s&"%tias '+ngit!"s* " a'"n H / ni!a!"s* n+ ,an si!+ t+!a$a "sta5'"%i!as "n "stas &+/"%%i+n"s1 L+s "%t+"s A / @ n+ "s n"%"sai+ " s"an %+n%"nt"s1 La %+nst%%i8n !"' !iagama !" "%t+"s* " ga "n 'a &at" !""%,a* s" ini%ia "n "' &nt+ %+n"n%i+na' O1 E' "%t+ A "m&i"a "n O* ta7n!+s" "n 'as !+s &+/"%%i+n"s &aa'"'as a "sta !i"%%i8n "n am5as1 A &ati !"' ";t"m+ A s" taa "' "%t+ @* n+ sin ant"s ,a5" t+ma!+ "n 'a &+/"%%i8n "ti%a' 'as s"is ni!a!"s !" '+ngit! "!a!"a "n "' s"gm"nt+ / "%t+ A1 E' "%t+ @ taa!+ "s +5'i%+ n+ a&a"%i"n!+ "n tama+ "a'* &+ "''+ / "m&'"an!+ "' m6t+!+ !" 'a ga ??-9 s" taa "st" "%t+ "n s '+ngit! "!a!"a t+man!+ 'as %in%+ ni!a!"s* !" s a'+ a5s+'t+* "n 'a !istan%ia a T5 A 1 Uni"n!+ "' &nt+ @ %+n "' O* &+ m"!i+ !" na '$n"a* t"n!"m+s 'a &+/"%%i8n "ti%a' / ,+i+nta' !"' "%t+ "s'tant"1
-. Dec%(p%ici#! de '! $ec"%r e! el epaci% Las %+m&+n"nt"s !" n "%t+ sita!+ "n "' "s&a%i+ s+n t"s* s"g3n 'as !i"%%i+n"s " si"n &aa !"t"mina "n g"n"a' a t+!+ %"&+* &!i6n!+s" !"s%+m&+n" "n t"s !i"%%i+n"s n+-%+&'ana"s " ti"an tam5i6n !i"%%i+n"s "s&"%$%as /a !"t"mina!as1 Un %as+ %+i"nt" "s "' " "m+s ";&"sa!+ "n 'a ga ??-??* "n "' "* "n (a)* "m+s 'as !+s &+/"%%i+n"s &in%i&a'"s* %+n "' ">" &+/"%ta!+ "n n &nt+* "n 'a &+/"%%i8n ,+i+nta'4 / "' ">" Y* tam5i6n &+/"%ta!+ "n n &nt+* "n 'a &+/"%%i8n "ti%a'1 En (5) a&a"%" "' &isma "%tang'a " s" #+ma "n "' "s&a%i+* %+n "' "%t+ A %+m+ !iag+na'* / 'as t"s &+/"%%i+n"s !"' mism+ s"g3n '+s t"s ">"s !"' sist"ma %++!"na!+1 Y "n (%) s" ti"n" !" n"+ 'as &+/"%%i+n"s &in%i&a'"s !"' mism+ &isma %+n "' "%t+ A / ss t"s %+m&+n"nt"s* na !" 'as %a'"s s" &+/"%ta "n n &nt+1 T+!+s '+s "%t+"s ti"n"n s +ig"n %+m3n "n "' &nt+ O1 Esta !"s%+m&+si%i8n !" n "%t+ %a'i"a* "n t"s sman!+s* s"g3n '+s ">"s %++!"na!+s "%tang'a"s* "s na +&"a%i8n m/ 3ti'* &+" &"miti7 ""m&'aa n "%t+ +5'i%+ &+ t"s "%t+"s !" '+ngit!"s "!a!"as1
. La /'er0a c%(% '!a (a!i"'d $ec"%rial La magnit! #"a &"!" s" !"ni!a* sim&'"m"nt"* + %+m+ im&'s+a + ta%t+a1 Una #"a a%t3a s+5" n %"&+ !"' m+!+ sigi"nt". (?) Cam5ian!+ 'a "'+%i!a! !"' %"&+* n+ ";isti"n!+ +ta " s" '" +&+nga. (2) %"an!+ +ta #"a !" "a%%i8n* + !" in"%ia* si "' %"&+ "st7 "n "&+s+ + %+n "'+%i!a! %+nstant"* /a " ti"n!" a " "' mism+ n+ %am5i" !" &+si%i8n4 () &+!%i"n!+ na !"#+ma%i8n "n "' "#"i!+ %"&+* " +igina +tas #"as int"nas1 N+s am+s a %+n%"ta a$ s+'am"nt"* a' s"gn!+ "#"%t+* a' !" 'as #"as " a%t3an s+5" n %"&+ " "st7 "n "&+s+* '+ " "n m"%7ni%a s" ''ama "st7ti%a1 Una #"a %a'i"a ti"n" t"s %aa%t"$sti%as* !" 'as " /a ,"m+s ,a5'a!+ "n '+s "%t+"s. <nt+ !" a&'i%a%i8n* !i"%%i8n-s"nti!+ / magnit!4 tam5i6n s" '"s ''ama magnit!* '$n"a !" a%%i8n / s"nti!+1 S+n 'as %aa%t"$sti%as g"n"a'"s " ,an !" t"n" '+s "%t+"s '+%a'ia!+s1
2. El pri!cipi% de la "ra!(ii#! de '!a /'er0a E' "#"%t+ ";t"n+ !" na #"a " a%t3a s+5" n %"&+ $gi!+ n+ s" a't"a si "' &nt+ !"
a&'i%a%i8n s" m"" sigi"n!+ 'a !i"%%i8n !" 'a '$n"a !" a%%i8n !" "sa #"a1 A "st" im&+tant" ,"%,+ s" '" %+n+%" %+n "' n+m5" !" &in%i&i+ !" 'a tansmisi8n* " "s m/ 3ti' &aa 'a s+'%i8n !" '+s &+5'"mas "'ati+s a 'as #"as1 La ga ??-?2 m"sta 'as a&'i%a%i+n"s " ti"n" "s" &in%i&i+ a n s"n%i''+ %as+ !" #"as %+&'ana"s1 Un %"&+ i"g'a s" '" s&+n" s+m"ti!+ a !+s #"as F ? Y F 2* a&'i%a!as "n '+s &nt+s A / @* %+m+ "m+s "n 'a #as" ?1 <+ "' &in%i&i+ "nm"a!+ "stas !+s #"as* !" s"nti!+ In"s+ / a&'i%a!as "n '+s &nt+s A / @* s" &"!"n %+nsi!"a a&'i%a!as "n 'as &+'+nga%i+n"s int"nas !" 'as !+s !i"%%i+n"s " s" "n%"ntan "n "' &nt+ C4 "s !"%i " %+n+%i"n!+ "' &nt+ !" a&'i%a%i8n !" "sas #"as magnit!"s* !i"%%i+n"s / s"nti!+* "!an "n "st" %as+ %+n%"nta!as "n "s" &nt+ C* 5astan!+ %+n &+n" "stas !+s #"as F ? Y F 2 s"g3n "m+s "n 'a #as" &aa t"n" 'a "s'tant" R1 O s"a* " 'a a%ta%i8n !" 'as !+s #"as ";t"nas F ? Y F 2 "n '+s &nt+s A / @* "s '+ mism+ " si a%tas" s+5" "' %"&+ na s+'a #"a "s'tant" R* a&'i%a!a "n %a'i" &nt+ !" 'a '$n"a DE-C1 Si"m&" "n 'a '$n"a ED1 Si "' &nt+ !" %+n%"n%ia #"a ina%%"si5'"* 'a #"a "s'tant" s" &"!" !"t"mina &+ "' m6t+!+ in!i%a!+ "n 'a ga ??-?1 En 'a #as" ? s" m"stan 'as !+s #"as !a!as %+&'ana"s* FI / F2 a&'i%a!as "n '+s &nt+s A / @* " %+n%"n "n n &nt+ ina%%"si5'"* a 'a ii"!a !" 'a ga1 Si a&'i%am+s "n A / @ !+s #"as F* iga'"s "n magnit!* "n 'a misma !i"%%i8n / s"nti!+ +&"st+* a '+ 'ag+ !" 'a '$n"a A@* %+m+ s" +5s"a "n 'a #as" * n+ s" a't"a "' "#"%t+ !" 'as #"as !a!as* F? Y F 2 a' n"ta'ias" "nt" s$ 'as !+s #"as F1 S" sman '+s "%t+"s F Y F ?* ""m&'a7n!+'"s &+ "' "%t+ R? * as$ %+m+ '+s "%t+"s F / F 2 &+ s sma R 21 Estas #"as R ? Y R 2 /a %+n%"n "n "' &nt+ C* %+m+ "m+s "n 'a #as" * &!i"n!+ smas" &aa +5t"n" 'a "s'tant" na' R* " &"!" %+nsi!"as" a&'i%a!a "n %a'i" &nt+ !" 'a '$n"a DE1 Las #"as iga'"s F &"!"n t"n" %a'i" magnit!* /a " "' &nt+ C "sta7 si"m&" "n 'a '$n"a DE1
3. Re'l"a!"e de $aria /'er0a c%!c'rre!"e Ca'i" sist"ma !" #"as %+n%"nt"s s" &"!" ""m&'aa &+ na s+'a #"a "s'tant"* " a%t3" "n "' &nt+ !" %+n%"n%ia1 En "' %as+ g"n"a' !" #"as n+-%+&'ana"s "st+s "%t+"s !"5"n "&"s"ntas" %+m+ "%t+"s !" #"as / ag"gas" a 'as &+/"%%i+n"s &in%i&a'"s1 En 'a ga ??-?B s" m"sta "' m6t+!+ !" !"t"mina%i8n !" 'a #"a "s'tant" &aa "st" %as+ g"n"a'1 EL DIAGRAMA DE 4UER5AS 1
Est" !iagama m"sta 'a !i"%%i8n " ti"n"n 'as t"s #"as " a%t3an "n "' &nt+ O4 &aa %+n"ni"n%ia !" "st" %as+* / "n g"n"a' !" t+!+s '+s &+5'"mas " a/am+s ";&+ni"n!+* 'a n+ta%i8n !" @+ (tam5i6n s" ''ama n+ta%i8n !" "ni%i) "s 'a " s" "m&'"a &aa !"signa 'as #"as "n s "nn%ia%i8n1 En "sta an+ta%i8n s" %+'+%a na '"ta ma/3s%'a a %a!a 'a!+ !" %a!a "%t+* "!an!+* &+ tant+* a'gnas "nt" "st+s "%t+"s* !"signan!+ %a!a "%t+ &+ 'as
!+s '"tas ma/3s%'as " ,a/ a ss !+s 'a!+s* "m&"an!+ &+ 'a ii"!a4 "s !"%i* s" '""n '+s "%t+"s !" ii"!a a !""%,a1 <"+ &aa "ita &+si5'"s %+n#si+n"s* si s" "m&'"aa "st" sist"ma !" n+ta%i8n "n 'as !+s &+/"%%i+n"s* s" "#"%t3a 3ni%am"nt" "n na !" "''as* "n "st" %as+ "n 'a &+/"%%i8n ,+i+nta' !" 'a ga ??-?B4 mi"ntas " "n 'a &+/"%%i8n "ti%a'* + "n +tas " #"an a!/a%"nt"s a 'a ,+i+nta' 'a s"i" !" "nm"a%i8n !" '+s "%t+"s s" ,a%" %+n 'as !+s '"tas s%"sias " in!i%an '+s !+s ";t"m+s !"' "%t+* " s" %+"s&+n!"n %+n 'as " gan "n 'a &+/"%%i8n ,+i+nta' a '+s !+s 'a!+s !" %a!a #"a* !" ii"!a a !""%,a1 L+s "%t+"s in!i%a!+s "n "st" !iagama !" #"as n+ &"%isan s" !i5>a!+s a "s%a'a* %+m+ s%"!" %+n "' !iagama !" "%t+"s1 EL DIAGRAMA DE VECTORES 1
Sigi"n!+ "' +!"n !" 'a n+ta%i8n !"' !iagama ant"i+* "st" !iagama !" "%t+"s* " ,a57 !" !i5>a'+ a "s%a'a* "m&i"a s %+nst%%i8n "n %a'i" &nt+ %+n"ni"nt" ta' %+m+ "' &nt+ A1 S" taa "' "%t+ A@ &aa'"'+ a 'a !i"%%i8n !" "sta #"a m+sta!a "n "' +t+ !iagama / s magnit! !" 0 i'+s s" t+ma s+5" "sta !i"%%i8n* sigi"n!+ "' m6t+!+ !" 'a ga ??-9* "!an!+ as$ '+%a'ia!+ "' &nt+ @1 <+ "st" mism+ &nt+ s" taa "' "%t+ @C* " ga "n '+ngit! "!a!"a "n 'a &+/"%%i8n "ti%a' t+man!+ "n "''a 'a magnit! !" B0 i'+s &aa as$ "!a '+%a'ia!+ "' &nt+ C1 <+ "st" ";t"m+ / &aa'"'am"nt" a 'as &+/"%%i+n"s !" 'a #"a CD s" taa7 "st" "%t+ "n "' " s" t+ma7 a "s%a'a na magnit! !" H0 i'+s* +5t"ni"n!+ as$ "' &nt+ D* 3'tim+ !" 'as t"s #"as " s" "st7n %+m&+ni"n!+1 N+ "!a7 /a m7s " ni D %+n A* &aa %+m&'"ta " ' &+'$g+n+ !" "%t+"s* si"n!+ "st" "%t+ AD "' "s'tant" " s" 5s%a5a* / s magnit! !" ?20 i'+s* "n 'a !i"%%i8n " ma%a 'a ga* n+s in!i%a7 " 'a a%%i8n !" '+s t"s "%t+"s* %+n ss #"as "s&"%tias* "ia'"n a 'a s+'a a%%i8n " ti"n" "' "%t+ "s'tant" in!i%a!+1
6. Re'l"a!"e de d% /'er0a paralela La "s'tant" !" !+s #"as " s"an &aa'"'as &"!" s" +ta s"n%i''a &aa'"'a* + tam5i6n &"!"n #+ma n &a !" #"as1 C+nsta* "' &a* !" !+s #"as " s"an iga'"s / &aa'"'as* &"+ !" s"nti!+s in"s+s4 "' "#"%t+ " &+!%" "st" &a a' a&'i%as" a n %"&+* "s "' !" +ta%i8n4 si 'a "s'tant" !" 'as !+s #"as &aa'"'as #"s" s+'am"nt" +ta t"%"a "%ta &aa'"'a* a a&'i%as" "sta "s'tant" s+5" n %"&+ n+ &+!%i$a nn%a 'a +ta%i8n !" mism+* /a " s&+n!$a "n "' %"&+ !i"sas a%%i+n"s* ta'"s %+m+ !"s&'aami"nt+* %+m&"si8n* "t%1 La "s'tant" !" !+s #"as &aa'"'as s" &"!" +5t"n" 7&i!am"nt" &+ % m6t+!+ in!i%a!+ "n 'a ga ??-?* "' %a' "s sim&'"m"nt" na a&'i%a%i8n "&"ti!a !"' &in%i&i+ in!i%a!+ "n 'a ga ??-?1 <"s"nt7n!+s" '+s t"s %as+s sigi"nt"s.
11 En 'a ga ??-?(a) !+s #"as !"siga'"s / &aa'"'as F? / F2 a%t3an "n 'a misma !i"%%i8n4 si s&+n"m+s " "n '+s &nt+s !" +ig"n !" %a!a na !" "''as* A / @* s" a&'i%an !+s #"as F iga'"s / CASO
%+ntaias "n s"nti!+* !"nt+ !" 'a misma !i"%%i8n* 5asta7 %+m&+n" 'as !+s &im"as #"as %ita!as %+n "sta #"a F* a '+ 'ag+ !" 'a '$n"a A@4 'as #"as !a!as &"!"n s" ""m&'aa!as &+ 'as !+s "s'tant"s R? Y R21 Estas "s'tant"s %+n%"n "n "' &nt+ C !an!+ a s " 'a "s'tant" t+ta' R* " "n "a'i!a! n+ !">a !" s" m7s " 'a sma aitm6ti%a !" 'as !+s #"as F ? / F2 !a!as1 N8t"s" " "n "st" %as+ 'a "s'tant" R s" "n%"nta "nt" 'as #"as !a!as1
&1 En 'a ga ??-?(5) s" &"s"ntan 'as !+s #"as !"siga'"s / &aa'"'as* F? / F2* %+m+ "n "' %as+ ant"i+* s8'+ " a,+a* an" ti"n"n 'a misma !i"%%i8n* ti"n"n s"nti!+s +&"st+s1 A&'i%an!+ 'a misma %+nst%%i8n " ant"i+m"nt" ,a5$am+s "#"%ta!+* 'as !+s "s'tant"s R? / R 2 %+n%"n "n n &nt+ C " a,+a "st7 m7s 5a>+ " "' &nt+ A* %+n na "s'tant" t+ta' RPF ?-F 2 * "n !i"%%i8n / s"nti!+ !" 'a #"a ma/+ " "s F ? 1 CASO
CASO 7 1
En 'a ga ??-?(%) 'as !+s #"as &aa'"'as F ? / F 2 s+n a,+a iga'"s "n magnit!* &"+ !" s"nti!+ %+ntai+* an t"ni"n!+ 'a misma !i"%%i8n %+m+ "n '+s %as+s ant"i+"s1 A' "#"%ta 'a misma %+m&+si%i8n !" #"as* %+m+ ant"s ,i%im+s* 'as !+s "s'tant"s R ? / R2 ti"n"n "' mism+ a'+ / a!"m7s s+n &aa'"'as* "s !"%i* " n+ %+n%"n %+m+ ant"s "n ning3n &nt+ C* !+n!" ,a5 $a " a&'i%as" 'a "s'tant"1 En "st" %as+ #+man '+ " s" ''ama n &a !" #"as* %/a a%%i8n ti"n!" a &+!%i n m+imi"nt+ giat+i+* %+n na a%%i8n "s'tant" " a'!7 F ?a P R ?51 L"g+ "' a'+ " ti"n" "st" &a !" #"as s" ''ama m+m"nt+ !"' &a* " "m+s a'" 6' a'+ !" na !" 'as #"as &+ 'a !istan%ia &"&"n!i%'a " ,a/ "nt" 'as !+s #"as1 Est" &a !" s"gm"nt+s #+man "n "a'i!a! na s&"%i" !iigi!a + 5i"%t+ " "s "' &aa'"'+gam+ #+ma!+ &+ "sas !+s #"as* / "' 7"a !" "st" &aa'"'+gam+ "s '+ " ant"s ,"m+s ''ama!+ "' m+m"nt + !"' &a * / 'a !i stan%ia JaK "s " ' 5a+ !"' &a1
18.
Re'l"a!"e de /'er0a paralela c%pla!are
A&'i%an!+ "' m6t+!+ !i%,+ !" 'a ga ??-? &+ a&'i%a%i+n"s s%"sias a %a'i" n3m"+ !" #"as &aa'"'as* %a'i"a " s"a s s"nti!+* &"!" "!a "!%i!+ si"m&"* "' %as+ " s"a* a na sim&'" #"a + a n &a !" #"as1 Sin "m5ag+* %an!+ "st7n im&'i%a!as m7s !" !+s + t"s #"as "s m7s &7%ti%+ "m&'"a "' m6t+!+ " am+s a " "n 'a ga ??-?H1 CASO 1. LA RESULTANTE 4INAL ES UNA 4UER5A 1
En 'a ga ??-?H(a) a&a"%"n %at+ #"as &aa'"'as* " n+s s+n !a!as "n n !iagama !" #"as* %+n +5>"t+ !" taa n !iagama !" "%t+"s &aa '+%a'ia 'a "s'tant"* a "s%a'a1 Em&'"an!+ 'a n+ta%i8n !" @+* 'as #"as "st7n !"signa!as &+ 'as '"tas A* @* C* "t%1* s+5" "' 'at"a' !" %a!a #"a* '"/6n!+s" !i%,as #"as !" ii"!a a !""%,a1 S" "m&i"a "' !iagama !" "%t+"s &+ %a'i" &nt+ A t+man!+ ,a%ia ai5a* "n
"s%a'a* "' a'+ !" 'a #"a A@ " a'" H g* t"ni6n!+s" "' &nt+ @1 A &ati !" "st" &nt+ s" t+ma ,a%ia a5a>+ 'a #"a @C* " a'" g* +5t"ni6n!+s" "' &nt+ C1 A &ati !" "st" &nt+ C / ,a%ia ai5a s" t+ma7 'a #"a CD &+ na magnit! !" g* t"ni6n!+s" "' &nt+ D4 a &ati !"' %a' s" t+ma7 'a 3'tima #"a DE* &+ n a'+ !" B g* / ,a%ia ai5a* &aa t"n" "' &nt+ na' E1 L"g+ 'a !istan%ia AE* s"7 'a "s'tant" 5s%a!a* " a'" g O5s6"s" "* 'as :"%,as !" 'as #"as ,an si!+ +miti!as &aa "st+s "%t+"s* /a " "' mism+ +!"n !" 'as '"tas a In!i%an!+ "' s"nti!+1
&. UN PAR DE 4UER5AS COMO RESULTANTE 1 En 'a ga ''-?H(5) "' mism+ sist"ma !" #"as t"n"m+s a,+a " 'as !a!as ant"i+m"nt"* ";%"&t+ " 'a #"a Q !" B gs "n " !" "sta !iigi!a ,a%ia ai5a a,+a '+ "st7 ,a%ia a5a>+1 En "' !iagama !" "%t+"s "' &nt+ E %+in%i!" %+n "' &nt+ A* !" a,$ " 'a #"a "s'tant" s"a %"+1 En "' &+'$g+n+ !" %"!as a / " s+n &aa'"'as* '"g+ 'a "s'tant" s"7 n &a !" #"as " a%t3an a '+ 'ag+ !" 'as %"!as a / "1 La magnit! !"' &a !" #"as "s !" * gs* s"g3n "s%a'a* !"s!" AO* %+m+ "m+s* "n "' !iagama !" "%t+"s1 E' s"nti!+ !" 'a #"a AO &"!" s" !"t"mina!+ CASO
%+nsi!"an!+ n ti7ng'+ "%t+ia'* "n "' %a' AO s"a n %+m&+n"nt"1 S" &"!" %+nsti "' ti7ng'+ int"i+ AO@ " m"sta %+n :"%,as t+!+s ss 'a!+s &aa in!i%a " 'a %+m&+n"nt" AO !"5" a%ta !"s!" A ,asta O1 E' &a "s'tant"* &+ '+ tant+* ti"n" na magnit!* n m+m"nt+* !" * ; B*H P B0* %m1 - g1 " a%t3a "n !i"%%i8n !" ii"!a a !""%,a1 Si 'as %"!as a / " %+in%i!i"an* "' &a "s'tant" s"$a %"+* / 'as #"as "sta$an "n "i'i5i+1
11.
Re'l"a!"e de /'er0a paralela !% c%pla!are
E' m6t+!+ ant"i+ !" &+'$g+n+s* !" %"!as / !" a!i+s* s" &"!" a&'i%a tam5i6n a #"as &aa'"'as " n+ s"an %+&'ana"s* &"+ 'a %+nst%%i8n ti"n" " s" "a'ia!a !+s "%"s* na &+ %a!a na !" 'as !+s &+/"%%i+n"s &in%i&a'"s a!/a%"nt"s* %+n +5>"t+ !" '+%a'ia 'a #"a "s'tant" "n "' "s&a%i+1
E EM+* / s"7n n"gatias %an!+ "stas #"as t"ngan " a%ta ,a%ia ai5a1 S OLUCIN En 'a ga ??-? "' +'m"n !" %a!a &at" ,a si!+ %a'%'a!+* %+m+ s" "* &aa "n%+nta n +'m"n t+ta' !" ? %m 1 E' %"nt+ !" ga"!a! !" '+s %"&+s " "s "' %"nt+ g"+m6ti%+ !" %a!a n+ !" "st+s &ismas* &!i"n!+ '+%a'ias" "st+s &nt+s taan!+ 'as !iag+na'"s " "m+s "n !i%,a ga1 + !" "sta &+/"%%i8n "ti%a' s" ,a taa!+ "' &+'$g+n+ !" a!i+s* "n %a'i" "s%a'a %+n"ni!a4 tam5i6n ,a taa!+ "' %+"s&+n!i"nt" &+'$g+n+ !" %"!as* "n 'as &+'+nga%i+n"s !" 'as '$n"as !" a%%i8n !" %a!a na !" 'as t"s #"as gaitat+ias %+m+ "n 'a ??-ga ??-?H(a) Las #"as "&"s"nta!as &+ 'as %"!as a / ! s" %+tan "n "' &nt+ M* '"g+ 'a "s'tant" R !"5" a%ta s"g3n 'a !i"%%i8n !" 'a '$n"a MN* "n 'a %a' "sta7 sita!+ "' %"nt+ !" ga"!a! g F1 En 'a &+/"%%i8n 'at"a' 'as mismas t"s #"as s" &"!"n " a%tan!+ "ti%a'm"nt" "n '+s &nt+s R * / @ / @ 1 E' &+'$g+n+ !" a!i+s "m&'"a!+ "n 'a &+/"%%i8n "ti%a' &"!" s" "m&'"a!+ a$* &"+ "' &+'$g+n+ !" %"!as ti"n" " s" taa!+ "n 'as &+'+nga%i+n"s %+"s&+n!i"nt"s a 'as '$n"as !" a%%i8n !" "sas t"s #"as !" ga"!a! "n "sa &+/"%%i8n 'at"a'1 O5s6"s" " 'a %"!a C n+ ,a si!+ taa!a &+" 'as #"as @C / CD ti"n"n 'a misma '$n"a !" a%%i8n "n "sta &+/"%%i8n1 Las %"!as a / ! s" %+tan "n "' &nt+ <* '"g+ 'a "s'tant" R &asa7 &+ "sta '$n"a <* / R t"n!7 tam5i6n " "sta sita!+ "n "sta '$n"a1 L"g+ 'as '$n"as MN / < !"t"minan 'a '$n"a !" a%%i8n !" "sta "s'tant" t+ta' R &"+ 'a a'ta !"' &nt+ G "s a3n !"s%+n+%i!a1 Si s&+n"m+s " '+s !i5>+s taa!+s "#"%t3an n gi+ !" 90W* &+!"m+s imagina "nt+n%"s " 'a &+/"%%i8n 'at"a' s" ,a %+n"ti!+ "n &+/"%%i8n "ti%a'* / " 'a &+/"%%i8n "ti%a' "!a tans#+ma!a "n &+/"%%i8n ,+i+nta'1 Es !"%i* i"n" a s" "st+ "ia'"nt" a " ,5i6am+s gia!+ "' +5>"t+ &aa &+n"'+ "n "sta n"a &+si%i8n1 Las #"as !" ga"!a! " ant"s s" !iig$an ,a%ia a5a>+ a,+a
a&a"%"7n !iigi!as ,a%ia 'a !""%,a1 E' & +'$g+n+ !" a!i+s s" "'" a taa !" n"+* &+ni"n!+ '+s "%t+"s &aa'"'+s a 'a n" a !i"%%i8n* / "' &+'$g+n+ !" %"!as s" taa "n &+'+nga%i8n !" 'as '$n"as !" a%%i8n !" "sas #"as gaitat+ias* a,+a ,a%ia 'a !""%,a1 Las %"!as a / ! s" %+tan "n "' &nt+ S* '+ " in!i%a " g ; ,a !" "sta sita!+ "n 'a '$n"a ST* / %+m+ tam5i6n t"n$a " "sta'+ "n 'a '$n"a <* "sta7 "n 'a int"s"%%i8n* %+n '+ %a' "!a '+%a'ia!+ "s" &nt+ !" a&'i%a%i8n gR* " !"t"mina7 'a &+si%i8n !"' &nt+ g FX "n 'a &+/"%%i8n "ti%a' (a,+a ,+i+nta')1 L"g+ "' &nt+ G "s "' %"nt+ !" ga"!a! 5s%a!+ !"' %"&+ !a!+* " "a n"st+ +5>"ti+1 E' m6t+!+ in!i%a!+ s" &"!" "m&'"a tam5i6n &aa "n%+nta "' %"nt+ !" ga"!a!* + %"nt+ sim&'"m"nt" !" %a'i" 7"a* "n %/+ %as+ '+s &+'$g+n+s !" '+s a!i+s / !" 'as %"!as n"%"sitan s" taa!+s s+'am"nt" !+s "%"s1
