❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❯♠❛ ■♥tr♦❞✉çã♦ à ❈✉r✈❛s P❧❛♥❛s
❆♥❞❡rs♦♥ ❞❡ ❆③❡✈❡❞♦ ●♦♠❡s
●♦✐â♥✐❛
❆♥❞❡rs♦♥ ❞❡ ❆③❡✈❡❞♦ ●♦♠❡s
❯♠❛ ■♥tr♦❞✉çã♦ à ❈✉r✈❛s P❧❛♥❛s
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛✉rí❧✐♦ ▼ár❝✐♦ ▼❡❧♦
Ficha catalográfica elaborada automaticamente
com os dados fornecidos pelo(a) autor(a), sob orientação do Sibi/UFG.
Gomes, Anderson de Azevedo
Uma Introdução à curvas planas [manuscrito] / Anderson de Azevedo Gomes. - 2015.
79 f.
Orientador: Prof. Dr. Maurílio Márcio Melo.
Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME) , Catalão, Programa de Pós-Graduação em Matemática (PROFMAT - profissional), Goiânia, 2015.
Inclui fotografias, gráfico, lista de figuras.
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❉❡❞✐❝♦ ❡ss❡ tr❛❜❛❧❤♦ ❛ ♠❡✉s ♣❛✐s✱ ❉❥❛❧♠❛ ●♦♠❡s ❞❛ ❙✐❧✈❛ ❡ ▼❛r✐❛ ❈❛r♠✐❧â♥❞✐❛ ❞❡ ❆③❡✈❡❞♦ q✉❡ ❝♦♥tr✐❜✉í✲ r❛♠ ❡ ❛♣♦✐❛r❛♠ s❡♠♣r❡ ♥❛ ♠✐♥❤❛ ❥♦r♥❛❞❛ ♣r♦✜ss✐♦♥❛❧✳ ➚ ♠✐♥❤❛ ❡s♣♦s❛ ❉✐♦♥❡ ❙✐❧✈❛ ❡ ❛♦ ♠❡✉ ✜❧❤♦ ●❛❜r✐❡❧ ❙✐❧✈❛ ●♦♠❡s q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ❡ ♠❡ ❝♦♥❢♦rt❛r❛♠ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❛♥❣úst✐❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ à ❉❡✉s ♣♦r ♥♦s ❞❛r ♦ ❞♦♠ ❞❛ ✈✐❞❛✱ ❛ s❛❜❡❞♦r✐❛ ❡ ❛ ✐♥✲ t❡❧✐❣ê♥❝✐❛✳ ➚ ❉❡✉s só ❛❣r❛❞❡ç♦✱ ♣♦✐s✱ ❊❧❡ s❛❜❡ ❞♦ q✉❡ ♣r❡❝✐s♦ ❡ ❞♦ q✉❡ s♦✉ ❝❛♣❛③ ❞❡ ❛❞♠✐♥✐str❛r✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ♣♦r ♠❡ ❛❥✉❞❛r❡♠ ❛ t♦❞♦ ♠♦♠❡♥t♦✳ ❊❧❡s ❝♦♥tr✐❜✉✐r❛♠ ♠✉✐t♦ ❝♦♠ ❝♦♥s❡❧❤♦s ✈❛❧✐♦s♦s✳ ▼❡ ♠♦str❛r❛♠ q✉❡ ❛tr❛✈és ❞❛ ♠✐♥❤❛ ❤♦♥❡st✐❞❛❞❡✱ tr❛✲ ❜❛❧❤♦ ❡ ❞❡❞✐❝❛çã♦ ♣♦❞❡♠♦s ❝♦♥s❡❣✉✐r ❞✐❛s ♠❡❧❤♦r❡s✳ ❉✐ss❡r❛♠ q✉❡✱ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦✱ ❝♦♥str✉✐♠♦s ✉♠ ♠✉♥❞♦ ♠❛✐s ❥✉st♦✱ ♦♥❞❡ ♦ ❤♦♠❡♠ ♣♦❞❡ ❞❡s❢r✉t❛r ♦ ❜❡♠✲❡st❛r ♣❡❧❛ ❡❞✉❝❛çã♦✳
❖❜r✐❣❛❞♦ ♣♦r s❡♠♣r❡ ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦✱ t♦♠❛♥❞♦ ♠✐♥❤❛s ❞✐✜❝✉❧❞❛❞❡s ❝♦♠♦ s❡♥❞♦ s✉❛s ❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛♥❞♦ ♣❛r❛ ♣♦❞❡r ✈❡♥❝ê✲❧❛s ❡ ❛❝✐♠❛ ❞❡ t✉❞♦ ♠❡ ♠♦str❛♥❞♦ q✉❡ ♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ ♠❛✐♦r é s❡❣✉✐r ❡ ❢❛③❡r ❛ ✈♦♥t❛❞❡ ❞❡ ❉❡✉s✳ ❚✉❞♦ ♦ q✉❡ ❝♦♥s❡❣✉✐ ❛té ❛q✉✐ ❞❡✈♦ ♣r✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s✱ ❞❡♣♦✐s ❛ ✈♦❝ês✳ ❆❣r❛❞❡ç♦✱ t❛♠❜é♠✱ ❛ t♦❞♦s ♦s ❢❛♠✐❧✐❛r❡s ❞❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ♣❡❧♦ ❛♣♦✐♦ r❡❝❡❜✐❞♦ ❞❡ ✈♦❝ês✳
❘❡s✉♠♦
❈♦♠❡ç♦ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ P❧❛♥♦ ❡ ❝♦♠♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ✉♠ ❱❡t♦r ♥♦ P❧❛♥♦✳ ▼♦str♦ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ✈❡t♦r❡s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❛ s♦♠❛ ❞❡ ✈❡t♦r❡s ❝♦♠ ❡①❡♠✲ ♣❧♦s ❡ ❝♦♥t✐♥✉♦ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ Pr♦❞✉t♦ ❊s❝❛❧❛r ❡ ❝♦♠♦ ❡♥❝♦♥tr❛r ♦ ➶♥❣✉❧♦ ❡♥tr❡ ♦s ✈❡t♦r❡s✳ ❉❡♣♦✐s✱ ❡♥❝♦♥tr❛♠♦s ❛ ♣r♦❥❡çã♦ ❞❡ ✈❡t♦r❡s ❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ✉♠ ♣♦♥t♦ ❡ ✉♠❛ r❡t❛✳
◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❢❛ç♦ ✉♠❛ r❡✈✐sã♦ s♦❜r❡ ❛s ❙❡çõ❡s ❈ô♥✐❝❛s✳ ❈♦♠❡ç♦ ❢❛❧❛♥❞♦ ❞❛ ❊❧✐♣s❡✱ ❞❡♣♦✐s ❞❛ ❍✐♣ér❜♦❧❡ ❡ ✜♥❛❧✐③♦ ❢❛❧❛♥❞♦ ❞❛ P❛rá❜♦❧❛ ❡ s✉❛s ✉t✐❧✐❞❛❞❡s✳
❊♠ s❡q✉ê♥❝✐❛✱ ♠♦str♦ ❛s ❊q✉❛çõ❡s P❛r❛♠étr✐❝❛s ❞❡ ✉♠❛ ❝✉r✈❛✳ ❈❤❡❣♦ ❛ ✉♠❛ ❢ór✲ ♠✉❧❛ ❞❡ ❝♦♠♦ ❡♥❝♦♥tr❛r ✉♠ ❱❡t♦r ❚❛♥❣❡♥t❡ ❡ ❝♦♠♦ ❡♥❝♦♥tr❛r ♦ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❆r❝♦✳ ▼♦str♦ ❛ ✉t✐❧✐❞❛❞❡ ❞❛s ❋ór♠✉❧❛s ❞❡ ❋r❡♥❡t ❡ ❢❛ç♦ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛s ❈✉r✈❛s P❧❛♥❛s ❡ s✉❛s ✉t✐❧✐❞❛❞❡s✳
❋✐♥❛❧✐③♦ ♠♦str❛♥❞♦ ❝♦♠♦ ❢❛③❡r ❛❧❣✉♥s ❣rá✜❝♦s ♥♦ ●❡♦●❡❜r❛ ❡ ❞❛ ■♠♣♦rtâ♥❝✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♥❛ ❋♦r♠❛çã♦ ❇ás✐❝❛ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛ ❞♦s ❙♦❢t✇❛r❡s ❞❡ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✳ ▼❡✉ ♦❜❥❡t✐✈♦ é ♠♦str❛r ❛ ❜❡❧❡③❛ ❞❛ ♠❛t❡♠át✐❝❛ ❡ ❛tr❛✐r ♣❡ss♦❛s ♣❛r❛ ❡ss❛ ár❡❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡
❱❡t♦r❡s✱ ❈ô♥✐❝❛s✱ ❈✉r✈❛s ❡ ❘❡❢❡r❡♥❝✐❛❧ ❞❡ ❋r❡♥❡t✳
❆❜str❛❝t
❇❡❣✐♥♥✐♥❣ ✇✐t❤ t❤❡ ❝♦♥❝❡♣t ♣❧❛♥ ❛♥❞ ❤♦✇ ✐t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❛ ❱❡❝t♦r P❧❛♥✳ ❙❤♦✇ t❤❡ ✐❞❡♥t✐t② ♦❢ ✈❡❝t♦rs ❛♥❞ t❤❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✈❡❝t♦r s✉♠ ✇✐t❤ ❡①❛♠✲ ♣❧❡s ❛♥❞ ❝♦♥t✐♥✉❡ ✇✐t❤ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❙❝❛❧❛r ♣r♦❞✉❝t ❛♥❞ ❤♦✇ t♦ ✜♥❞ t❤❡ ❆♥❣❧❡ ❇❡t✇❡❡♥ ❱❡❝t♦rs✳ ❆❢t❡r ✇❡ ✜♥❞ t❤❡ ♣r♦❥❡❝t✐♦♥ ✈❡❝t♦rs ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛ ♣♦✐♥t ❛♥❞ ❛ ❙tr❛✐❣❤t✳
■♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✱ ■ r❡✈✐❡✇ ♦♥ ❈♦♥✐❝ ❙❡❝t✐♦♥s✳ ❚r❛❝t ♦❢ ❊❧❧✐♣s❡✱ ❛❢t❡r ❍②♣❡r❜♦❧❡ ❛♥❞ ❛♥❛❧②③❡ t❛❧❦✐♥❣ ❛❜♦✉t t❤❡ ♣❛r❛❜❧❡ ❛♥❞ ✐ts ✉t✐❧✐t✐❡s✳
■♥ s❡q✉❡♥❝❡✱ s❤♦✇ t❤❡ P❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥s ♦❢ ❛ ❝✉r✈❡✳ ❈♦♠❡ t♦ ❛ ❢♦r♠✉❧❛ ♦❢ ❤♦✇ t♦ ✜♥❞ ❛ t❛♥❣❡♥t ✈❡❝t♦r ❛♥❞ ❤♦✇ t♦ ✜♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ ❛♥ ❛r❝✳ ❙❤♦✇ t❤❡ ✉s❡❢✉❧♥❡ss ♦❢ t❤❡ ❋r❡♥❡t ❢♦r♠✉❧❛s ❛♥❞ ❞♦ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ P❧❛♥❡ ❈✉r✈❡s ❛♥❞ t❤❡✐r ✉t✐❧✐t✐❡s✳
❚♦ ❡♥❞ tr❛❝t ■♠♣♦rt❛♥❝❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ✐♥ Pr✐♠❛r② ❊❞✉❝❛t✐♦♥ ✇✐t❤ ❛ ❧✐tt❧❡ ❤✐st♦r② ♦❢ ❉②♥❛♠✐❝ ●❡♦♠❡tr② ❙♦❢t✇❛r❡ ❛♥❞ ❙♦❢t✇❛r❡ ●❡♦❣❡❜r❛✱ s❤♦✇✐♥❣ ❤♦✇ t♦ ♠❛❦❡ s♦♠❡ ❝❤❛rts ✐♥ ●❡♦●❡❜r❛✳ ▼② ❣♦❛❧ ✐s t♦ s❤♦✇ t❤❡ ❜❡❛✉t② ♦❢ ♠❛t❤❡♠❛t✐❝s ❛♥❞ ❛ttr❛❝t ♣❡♦♣❧❡ t♦ t❤❡ ❛r❡❛✳
❑❡②✇♦r❞s
❱❡❝t♦rs✱ ❈♦♥✐❝✱ ❈✉r✈❡s ❛♥❞ ❋r❡♥❡t ❋♦r♠✉❧❛s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ P♦♥t♦ ❡♠ ✉♠ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✹ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❙❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✺ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❋♦rç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✻ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✼ ❘❡♣r❡s❡♥t❛çã♦ ●rá✜❝❛ ❞❡ ✉♠ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✽ ❖ ❖♣♦st♦ ❞❡ ✉♠ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✾ ❆ ❙♦♠❛ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✵ ❱❡t♦r ♥❛ ❇❛s❡ ❈❛♥ô♥✐❝❛ ❡♠ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✶✶ ➶♥❣✉❧♦ ❡♥tr❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✷ ❱❡t♦r ❉❡s❧♦❝❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✸ ❱❡t♦r ❉❡s❧♦❝❛♠❡♥t♦ ❚♦t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✹ ❱❡❧♦❝✐❞❛❞❡ ❊s❝❛❧❛r ❞❡ ✉♠ ❆✈✐ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✺ ❋♦rç❛ ❘❡s✉❧t❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✻ Pr♦❥❡çã♦ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✼ ❖ ➶♥❣✉❧♦ ❊♥tr❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✽ P❡r♣❡♥❞✐❝✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✾ ❉✐stâ♥❝✐❛ ❞❡ ✉♠ P♦♥t♦ P0 à ✉♠❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷✵ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✶ ❋♦♥t❡ ▲✉♠✐♥♦s❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✷ ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✸ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✹ ❘❛í③❡s ❞❡ ✉♠❛ ❊q✉❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✺ ❖ P♦♥t♦ ❞❛ P❛rá❜♦❧❛ ❡stá ❆❝✐♠❛ ❞♦ P♦♥t♦ ❞❛ ❘❡t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✻ ■♥❝❧✐♥❛çã♦ ❞❛ ❘❡t❛F Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✷✼ ❘❡t❛s P❡r♣❡♥❞✐❝✉❧❛r❡s q✉❛♥❞♦aa′
=−1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✽ ❘❡t❛ ◆♦r♠❛❧ à P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✾ ❚r❛ç♦s ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✵ ▲✐♠❛ç♦♥ ❞❡ P❛s❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✶ ❈✐ssó✐❞❡ ❞❡ ❉✐♦❝❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✷ ❚r❛ç♦s ❙❡♠❡❧❤❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✸ ❊s♣✐r❛❧ ▲♦❣❛rít♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✸✹ P❛rá❜♦❧❛ ❡ ❈ír❝✉❧♦ ❖s❝✉❧❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✺ ●rá✜❝♦ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✻ ●rá✜❝♦ ❞❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✼ ❈✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸✽ ❍✐♣♦❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✾ ❆stró✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✵ ❈❛r❞✐ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✶ ❋✐❣✉r❛ ❙❡♠❡❧❤❛♥t❡ ❛ ✉♠❛ ❇♦r❜♦❧❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✹
✷ ❖ P❧❛♥♦ ✶✻
✷✳✶ ❱❡t♦r❡s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ Pr♦❞✉t♦ ❊s❝❛❧❛r ❡ ➶♥❣✉❧♦ ❡♥tr❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✶ ❱❡t♦r ❉❡s❧♦❝❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✷ ❱❡t♦r ❘❡s✉❧t❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✺ Pr♦❥❡çã♦ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✻ ❊q✉❛çã♦ ❞❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸ ❙❡çõ❡s ❈ô♥✐❝❛s ✸✸
✸✳✶ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹ ❊q✉❛çõ❡s P❛r❛♠étr✐❝❛s ❞❡ ✉♠❛ ❈✉r✈❛ ✹✽
✹✳✶ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ ▼✉❞❛♥ç❛ ❞❡ P❛râ♠❡tr♦ ❡ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ❆r❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷✳✶ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✸ ❚❡♦r✐❛ ▲♦❝❛❧ ❞❛s ❈✉r✈❛s P❧❛♥❛s ❡ ❋ór♠✉❧❛s ❞❡ ❋r❡♥❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✹ ❊①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✺ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛s ❈✉r✈❛s P❧❛♥❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✻ ▼♦✈✐♠❡♥t♦s P❧❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✺ ❯t✐❧✐③❛♥❞♦ ♦ ●❡♦❣❡❜r❛ ✻✹
✶ ■♥tr♦❞✉çã♦
◆❛ ♠✐♥❤❛ ❡①♣❡r✐ê♥❝✐❛✱ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ♣❡r❝❡❜✐ q✉❡ ♥ã♦ ❜❛st❛ ♦❢❡✲ r❡❝❡r ❝♦♥❤❡❝✐♠❡♥t♦✱ é ♣r❡❝✐s♦ ♦❢❡rt❛r ✉♠❛ ❡❞✉❝❛çã♦ q✉❡ ❛t❡♥❞❛ às ♥❡❝❡ss✐❞❛❞❡s ❞❡ ❢♦r♠❛çã♦ ❞♦ ❛❧✉♥♦ ❝♦♠♦ s❡r s♦❝✐❛❧ ❝rít✐❝♦ ❡ ❛♣t♦ ❛ ❛❣✐r ♥♦ ❛♠❜✐❡♥t❡ s♦❝✐❛❧ ❡♠ q✉❡ ❡st❡ ❡stá ✐♥s❡r✐❞♦✱ ♦✉ s❡❥❛✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ s✉❛ r❡❛❧✐❞❛❞❡✳ P❡r❝❡❜❡✲s❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ ♣❛rt✐❝✐♣❛çã♦ ♠❛✐s ❡❢❡t✐✈❛ ❞❛ ❢❛♠í❧✐❛ ♥❛ ❡❞✉❝❛çã♦ ❡s❝♦❧❛r✳ ◆♦ ❡♥t❛♥t♦✱ s❡ t♦r♥❛ ♥❡❝❡ssá✲ r✐♦ ❞❡s♣❡rt❛r ♦ ❞❡s❡❥♦ ❞❡ ♠✉❞❛♥ç❛ ❞❡ ♣♦st✉r❛ ♣❡ss♦❛❧ ❡ s♦❝✐♦❝✉❧t✉r❛❧✱ ✐♠♣❧❛♥t❛r ♥♦✈♦s ♣❛❞rõ❡s só❝✐♦✲♣♦❧ít✐❝♦s✱ ♦♣♦rt✉♥✐③❛r ♦ ❞✐á❧♦❣♦ tã♦ ♥❡❝❡ssár✐♦ ❛♦ ❡①❡r❝í❝✐♦ ❞❛ ❞❡♠♦❝r❛❝✐❛ ❡ ❜✉s❝❛r ♦ r❡s❣❛t❡ ❞❡ ✈❛❧♦r❡s ❢❛♠✐❧✐❛r❡s✱ ♠♦r❛✐s ❡ ❝r✐stã♦s✱ ❜❡♠ ❝♦♠♦ ✉♠ ✐♥✈❡st✐♠❡♥t♦ ♠❛✐♦r ❡♠ ♣♦❧ít✐❝❛s ♣ú❜❧✐❝❛s ♣❛r❛ ❛ ❊❞✉❝❛çã♦ ✈✐s❛♥❞♦ ❛ ♠❡❧❤♦r✐❛ ❝♦♥tí♥✉❛ ❞❛ ❡str✉t✉r❛ ❞❡ ❡♥s✐♥♦✳
❆❧✐❛❞❛ ❛ ❡ss❛s ✐♠♣♦rt❛♥t❡s ♥❡❝❡ss✐❞❛❞❡s ❡stá ♦ ❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♣r♦❢❡ss♦r✱ ♣r✐♥❝✐✲ ♣❛❧♠❡♥t❡ q✉❛♥t♦ às ♠❡t♦❞♦❧♦❣✐❛s ❞❡ ❡♥s✐♥♦ ✉t✐❧✐③❛❞❛s✱ q✉❡ ♣r❡❝✐s❛♠ s❡r r❡❛✈❛❧✐❛❞❛s ❡ ♦s r❡s✉❧t❛❞♦s r❡♣❡♥s❛❞♦s✳ ❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❛ ❡❞✉❝❛çã♦ é ♦ ❞❡ ❧❡✈❛r ♦ ❛❧✉♥♦ ❝♦♠ ❝❡rt♦ ♥í✈❡❧ ✐♥✐❝✐❛❧ ❛ ❛t✐♥❣✐r ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥í✈❡❧ ✜♥❛❧✳ ❙❡ ❝♦♥s❡❣✉✐r ❢❛③❡r ❝♦♠ q✉❡ ♦ ❛❧✉♥♦ ♣❛ss❡ ❞❡ ✉♠ ♥í✈❡❧ ♣❛r❛ ♦✉tr♦✱ ❡♥tã♦ r❡❣✐str❛♠♦s ✉♠ ♣r♦❝❡ss♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✳ ❈❛❜❡ ❛♦s ❡❞✉❝❛❞♦r❡s ♣r♦♣♦r❝✐♦♥❛r s✐t✉❛çõ❡s ❞❡ ✐♥t❡r❛çã♦✱ t❛✐s q✉❡✱ ❞❡s♣❡rt❡♠ ♥♦ ❡❞✉✲ ❝❛♥❞♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ♦❜❥❡t♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❝♦♠ s❡✉s ❝♦❧❡❣❛s ❡ ❝♦♠ ♦s ♣ró♣r✐♦s ♣r♦❢❡ss♦r❡s✳ ❖s r❡❝✉rs♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ❜❡♠ ❝♦♠♦ ❛s ♥♦✈❛s ❢❡rr❛♠❡♥t❛s ❞❡ tr❛❜❛❧❤♦ ❝♦♠♦ é ♦ ❝❛s♦ ❞♦s s♦❢t✇❛r❡s✱ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ❞❡ t♦r♥❛r ♦ tr❛❜❛❧❤♦ ❡♠ s❛❧❛ ♠❛✐s ❞✐♥â♠✐❝♦ ❡ q✉❡ ♣♦ss✐❜✐❧✐t❡♠ ❛♦ ❛❧✉♥♦ s❡ ❛♣r♦♣r✐❛r ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❡❢❡t✐✈❛ ❞♦s ❝♦♥t❡ú❞♦s tr❛❜❛❧❤❛❞♦s✳ P♦rq✉❡✱ ♠❡s♠♦ q✉❡ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ♦❝♦rr❛ ♥❛ ✐♥t✐♠✐❞❛❞❡ ❞♦ s✉❥❡✐t♦✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦ ❞á✲s❡ ♥❛ ❞✐✈❡rs✐❞❛❞❡ ❡ ♥❛ q✉❛❧✐❞❛❞❡ ❞❛s s✉❛s ✐♥t❡r❛çõ❡s✳
P♦r ✐ss♦✱ ❛ ❛çã♦ ❡❞✉❝❛t✐✈❛ ❞❛ ❡s❝♦❧❛ ❞❡✈❡ ♣r♦♣✐❝✐❛r ❛♦ ❛❧✉♥♦ ♦♣♦rt✉♥✐❞❛❞❡s ♣❛r❛ q✉❡ ❡ss❡ s❡❥❛ ✐♥❞✉③✐❞♦ ❛ ✉♠ ❡s❢♦rç♦ ✐♥t❡♥❝✐♦♥❛❧✱ ✈✐s❛♥❞♦ r❡s✉❧t❛❞♦s ❡s♣❡r❛❞♦s ❡ ❝♦♠✲ ♣r❡❡♥❞✐❞♦s✳ ❆ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ♥ã♦ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❜❛♥❛❧✐③❛r ♦s ❝♦♥t❡ú❞♦s✱ ♠❛s ❝♦♠♦ r❡❝✉rs♦ ♣❡❞❛❣ó❣✐❝♦✳ ❊♥t❡♥❞❡♥❞♦ q✉❡ s❡ tr❛t❛✱ ♣♦rt❛♥t♦✱ ❞❡ ✉♠❛ ♣♦♥t❡ ❡♥tr❡ ❛ t❡♦r✐❛ ❡ ❛ ♣rát✐❝❛✱ ❡♥tr❡ ♦ ❝♦♥❝❡✐t♦ ❡ ❛ ✈✐✈ê♥❝✐❛✳
❖ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ ❞❡ ♠♦t✐✈❛çã♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❢❛t♦r ♣s✐❝♦❧ó❣✐❝♦ ♣♦❞❡ s❡r ❝♦♥✲ s✐❞❡r❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢❛t♦r❡s✱ ♦✉ ✉♠ ♣r♦❝❡ss♦✱ ♦✉ s❡❥❛✱ ✉♠❛ ❞❡❝✐sã♦ q✉❡ ♣❛rt❡ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡ss♦❛ ❡♠ ❜✉s❝❛ ❞❡ ✉♠ ♦❜❥❡t✐✈♦ ❛ s❡r ❞❡✜♥✐❞♦✱ ❡ ✈❡r ♦ ♠❡s♠♦ ❝♦♠♦ ✉♠ ❞❡s❛✜♦ ❛ s❡r ❞✐r❡❝✐♦♥❛❞♦✳ ❖ ♣r♦❢❡ss♦r ❛♦ ✐♥❝❡♥t✐✈❛r ♦s ❛❧✉♥♦s✱ ❥♦❣❛ ❝♦♠ ❡♠♦çã♦ ❞❛♥❞♦ ❡①❡♠♣❧♦s ❞♦ q✉❡ ❥á ♦❝♦rr❡✉ ❡♠ s✉❛ ✈✐❞❛✱ ♦✉ q✉❡ ♣♦❞❡ ♦❝♦rr❡r ♥❛ ✈✐❞❛ ❞❡st❡s ❛❧✉♥♦s ✉s❛♥❞♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❞♦ s❡♥s♦ ❝♦♠✉♠ q✉❡ ♣♦ss❛ ❝❛t✐✈á✲❧♦s✳ ◗✉❡ ♦ ♣r♦❢❡ss♦r ♣♦ss❛ ❡st❛r ❛t❡♥t♦ ❛♦ s❡✉ r❡❛❧ ♣❛♣❡❧✱ ❞❡ q✉❡ ❞❡✈❡ ❝✉♠♣r✐r ♥❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ r❡❧❛✲ ❝✐♦♥❛♠❡♥t♦ ❝♦♠ ♦s ❛❧✉♥♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ♠♦t✐✈❛çã♦✱ ✐♥❝❡♥t✐✈❛♥❞♦✲♦s ❡♠ t❛r❡❢❛s ♣♦s✐t✐✈❛s✱ ❞❡s❛✜❛♥❞♦✲♦s ♠❡s♠♦ q✉❡ s❡❥❛ ❞✐❢í❝✐❧✳
❍á ❞♦✐s t✐♣♦s ❞❡ ♠♦t✐✈❛çã♦ q✉❡ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ tr❛❜❛❧❤❛r✿ ❛ ✐♥trí♥s❡❝❛ ♦✉ ❛ ❡①trí♥✲ s❡❝❛✳ ❆ ♠♦t✐✈❛çã♦ ✐♥trí♥s❡❝❛ r❡❢❡r❡✲s❡ à ❡s❝♦❧❤❛ ❡ r❡❛❧✐③❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞❛ ❛t✐t✉❞❡ ♣♦r s✉❛ ♣ró♣r✐❛ ❝❛✉s❛✱ ♦✉ ♣♦r ✐♥t❡r❡ss❡ ❡ s❛t✐s❢❛çã♦✳ ❏á ♠♦t✐✈❛çã♦ ❡①trí♥s❡❝❛ é ❞❡✜♥✐❞❛ ♣❛r❛ tr❛❜❛❧❤❛r✱ ♦✉ s❡❥❛✱ ❡♠ ❜✉s❝❛ ❞❡ r❡❝♦♠♣❡♥s❛s ♠❛t❡r✐❛✐s ♥❛ s♦❝✐❡❞❛❞❡✳ ❆ r❡❧❛çã♦ ❞❡ ♣r♦❢❡ss♦r ❡ ❛❧✉♥♦ ❞❡✈❡✲s❡ ✐♥t❡r❛❣✐r ❝❛❞❛ ✈❡③ ♠❛✐s✱ ♥❛ ❝♦♥✜❛♥ç❛ ❞❛ r❡❧❡✈â♥❝✐❛ ❞❡ ♣r♦♠♦✈❡r ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ✉♠❛ r❡❧❛çã♦ s❡❣✉r❛ q✉❡ ♣♦ss❛♠ ❡♥✈♦❧✈❡r ♦s ♣❛✐s ❡ ♣r♦❢❡ss♦✲ r❡s✱ ❝♦♥❤❡❝❡♥❞♦ ❛ss✐♠ ♦s s✉❝❡ss♦s ❡ ♦s ❢r❛❝❛ss♦s ✭r❡♣r♦✈❛çõ❡s✮ ❝♦♠ ❛♣r❡♥❞✐③❛❣❡♠✱ q✉❡ ♦❝♦rr❡♠ ♥❛ ✈✐❞❛ ❡st✉❞❛♥t✐❧ ❞❡ ❝❛❞❛ ❛❧✉♥♦✳
❊♠ ❜✉s❝❛ ❞❛ ♠♦t✐✈❛çã♦✱ ❡❧❛ s❡ ❞á ♥❛ ♦r❣❛♥✐③❛çã♦ ❞❛ ❡s❝♦❧❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❛ s❛❧❛ ❞❡ ❛✉❧❛✱ ❝♦♠ ❛❧❣✉♥s ♦❜❥❡t✐✈♦s✿
✲ ♣r♦♠♦✈❡r ♦♣♦rt✉♥✐❞❛❞❡s ♣❛r❛ q✉❡ s❡❥❛♠ r❡❝♦♥❤❡❝✐❞❛s ♣❡❧❛ ❛♣r❡♥❞✐③❛❣❡♠❀ ✲ ♣r♦♣♦r❝✐♦♥❛r ❧✐❜❡r❞❛❞❡ ♣❛r❛ ♦s ❛❧✉♥♦s ❢❛③❡r❡♠ ❡s❝♦❧❤❛s❀
✲ ♣r♦♠♦✈❡r ❛♠♣❧❛ ✐♥t❡r❛çã♦ s♦❝✐❛❧ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❞❡ ❤❛❜✐❧✐❞❛❞❡s s♦❝✐❛✐s❀ ✲ ❛✉♠❡♥t❛r ❛ ❛tr❛çã♦ ✐♥trí♥s❡❝❛✳
❆ ✐♥t❡❧✐❣ê♥❝✐❛ ❡①❡r❝❡ ❣r❛♥❞❡ ♣❛rt❡ ♥❛ ✐♥✢✉ê♥❝✐❛ ❞❛ ♠♦t✐✈❛çã♦ ❞♦ ❛♣r❡♥❞❡r✱ ❡ ♦s ♠❡❝❛♥✐s♠♦s ♣s✐❝♦❧ó❣✐❝♦s ❞❛ ♠♦t✐✈❛çã♦ ❞♦ ❛❧✉♥♦ s❡ ❝♦♠♣õ❡♠ ♣♦r ❢❛t♦r❡s ❝♦♠♦ ❝r❡♥ç❛s ❞❡ ❛✉t♦ ❡✜❝á❝✐❛✱ ♦♥❞❡ ♦ ♥í✈❡❧ ❞❡ ♠♦t✐✈❛çã♦ ❞❡ ✉♠❛ ♣❡ss♦❛ é ❞❡t❡r♠✐♥❛❞♦ ❡♠ ❢✉♥çã♦ ❞♦s ❥✉❧❣❛♠❡♥t♦s q✉❡ tê♠ ✉♠ ✐♥❝❡♥t✐✈♦ ❞❡ ❛❣✐r ❡ ✐♥❞✐❝❛r ✉♠❛ ❞✐r❡çã♦ às s✉❛s ❛çõ❡s ❛♥t❡❝✐♣❛❞❛s ♥❛ ♠❡♥t❡ ♣♦❞❡♥❞♦ s❡r r❡❛❧✐③❛❞❛s ❡ ♦❜t❡r ❣r❛♥❞❡s r❡s✉❧t❛❞♦s✳
◆♦ ❡♥t❛♥t♦✱ ❛ ♠♦t✐✈❛çã♦ é ❡ss❡♥❝✐❛❧ ♥❛ ✈✐❞❛ ❞❡ ❝❛❞❛ ✐♥❞✐✈í❞✉♦ ♣❛r❛ q✉❡ ❡❧❡ ♣♦ss❛ t❡r ❣r❛♥❞❡s s✉❝❡ss♦s t❛♥t♦ ♥❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✱ ❝♦♠♦ ♥❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧ ✭s♦❝✐❛❧✮✱ ❜✉s❝❛♥❞♦ ❛ ❛❧t♦✲❡st✐♠❛ ❛ ❝❛❞❛ ❞✐❛✱ ♠❡s♠♦ ❝♦♠ ❛s ❞✐✜❝✉❧❞❛❞❡s q✉❡ s❡ ❡♥❝♦♥tr❛ ♥♦ ♠✉♥❞♦ ❞❡ ❤♦❥❡✳
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ❝❛r❛❝t❡ríst✐❝❛ ♠❛✐♦r ❛♠♣❧✐❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❛❜r✐♥❞♦ ❛s✲ s✐♠ ♥♦✈❛s ♣❡rs♣❡❝t✐✈❛s ❞❡ tr❛❜❛❧❤♦✱ ❝♦♥❝r❡t✐③❛♥❞♦ ❝♦♠ ✐ss♦ ❛ ♥♦ss❛ ❝♦♥tr✐❜✉✐çã♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♥♦ss❛ s♦❝✐❡❞❛❞❡✳ P❛r❛ t❛❧✱ ❢♦✐ ❛♣♦✐❛❞♦ ❡♠ r❡❢❡r❡♥❝✐❛✐s t❡ór✐❝♦s q✉❡ ❝♦♥tr✐❜✉íss❡♠ ♣❛r❛ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ r❡✐t❡r❛♥❞♦ q✉❡ ❛ ❛♥á❧✐s❡✱ ❛ ♣❡sq✉✐s❛ ❡ ♦ ❡st✉❞♦ ❢♦r♠❛♠ ♦ ♣r✐♥❝✐♣❛❧ ♠❡✐♦ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞♦ t❡♠❛✳
❆s ❢✉♥çõ❡s sã♦ ✉t✐❧✐③❛❞❛s ❡♠ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❝á❧❝✉❧♦s r♦t✐♥❡✐r♦s ❝♦♠♦ ❥✉r♦s✱ ♣r♦❞✉t✐✈✐❞❛❞❡ ❞❡ ✉♠❛ ❡♠♣r❡s❛✱ ❡t❝✳ P♦❞❡♠ s❡r ❡①♣r❡ss❛s ❣r❛✜❝❛♠❡♥t❡✱ ♦ q✉❡ ❢❛❝✐❧✐t❛ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦ ❝á❧❝✉❧♦✳ P♦r ❡ss❡ ♠♦t✐✈♦✱ ❞❡s❡♥✈♦❧✈♦ ❡ss❡ tr❛❜❛❧❤♦ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛r ❛♦ ❛❧✉♥♦ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s r❡❢❡r❡♥t❡s às ❢✉♥çõ❡s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❝❡♥tr❛❧ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ ✉♠❛ ♣r♦♣♦st❛ ♠❡✲ t♦❞♦❧ó❣✐❝❛ q✉❡ ✉t✐❧✐③❛ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ❝♦♠♦ r❡❝✉rs♦ ♣❡❞❛❣ó❣✐❝♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ❢✉♥çõ❡s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✱ ❜❡♠ ❝♦♠♦ ♦ ❡st✉❞♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❖ tr❛❜❛❧❤♦ ❡stá ❢✉♥❞❛♠❡♥t❛❞♦ ♥❛s ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s ❡❞✉❝❛❝✐✲ ♦♥❛✐s✱ t❡❝♥♦❧♦❣✐❛s ❞❛ ✐♥❢♦r♠❛çã♦ ❡ ❝♦♠✉♥✐❝❛çã♦✱ ♥❛ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛ q✉❡ ✈ê♠ ♣r❛ ✜❝❛r✱ ❝♦♠ ✉♠ s✉❝❡ss♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧✳
P♦r ✜♠✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s s♦❜r❡ ❛ ♣r♦♣♦st❛ ♣❡❞❛❣ó❣✐❝❛ ❞❡ss❡ tr❛❜❛❧❤♦✳
✷ ❖ P❧❛♥♦
◆❡st❛ s❡çã♦ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❧❛♥♦✱ ✈❡t♦r❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ❛♣❧✐❝❛✲ çõ❡s✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✐♥❞✐❝❛♠♦s ❬✷❪✱ ❬✻❪✱ ❬✼❪ ❡ ❬✽❪✳
❈♦♥s✐❞❡r❡ ♦ ♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ♣❛r ❞❡ r❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s x ❡ y✳ ❙❡❥❛ P ✉♠
♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦✳ P❡❧♦ ♣♦♥t♦P ♣♦❞❡♠♦s ♣❛ss❛r ✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛ à r❡t❛ xq✉❡ é x′ ❡ ✉♠❛ ú♥✐❝❛ ♣❛r❛❧❡❧❛ à r❡t❛
yq✉❡ é y′✳ ❆ ✐♥t❡rs❡çã♦ ❞❡st❛s r❡t❛s ♣❛r❛❧❡❧❛s é ♦
♣♦♥t♦P✳ ❖s ♥ú♠❡r♦sa ❡b sã♦ ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r ❛❜s❝✐ss❛ ❡ ♦r❞❡♥❛❞❛
❞♦ ♣♦♥t♦ P q✉❡ ❝♦♥st✐t✉❡♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ P✳ P❛r❛ ✐♥❞✐❝❛r q✉❡ ♦ ♣♦♥t♦ P t❡♠
❛❜s❝✐ss❛a ❡ ♦r❞❡♥❛❞❛ b ✉s❛♠♦s ❛ ♥♦t❛çã♦ P(a, b)✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✶✳
❋✐❣✉r❛ ✶✿ P♦♥t♦ ❡♠ ✉♠ P❧❛♥♦
P❡❧❛ ❝♦♥str✉çã♦✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♣♦♥✲ t♦s ❞♦ ♣❧❛♥♦ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s(a, b)✳ P❛r❛ ❝❛❧❝✉❧❛r♠♦s ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s P(a, b) ❡ Q(c, d)✱ ✉s❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❡ ❡♥✲ ❝♦♥tr❛♠♦s q✉❡ d(P, Q) = p(c−a)2+ (d−b)2✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✷✳
❋✐❣✉r❛ ✷✿ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s
✷✳✶ ❱❡t♦r❡s ♥♦ P❧❛♥♦
◗✉❛♥t♦ ❛♦ ♣♦♥t♦ P(a, b) 6= (0,0)✱ ♣♦❞❡♠♦s ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r ❛ ❡st❡ ♣♦♥t♦ ✉♠❛ s❡t❛ ❝♦♠ ♦r✐❣❡♠ ❡♠ (0,0) ❡ ❛ ♦✉tr❛ ❡①tr❡♠✐❞❛❞❡ ❡♠ P✳ ❯♠ ♣❛r ♦r❞❡♥❛❞♦ ♣♦❞❡ s❡r
r❡♣r❡s❡♥t❛❞♦ ❣r❛✜❝❛♠❡♥t❡ ♣♦r ✉♠ ♣♦♥t♦ ♦✉ ♣♦r ✉♠❛ s❡t❛✳ ◗✉❛♥❞♦ ✉t✐❧✐③❛♠♦s s❡t❛✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ❡st❡ ♣❛r ♦r❞❡♥❛❞♦ ❞✐r❡çã♦✱ s❡♥t✐❞♦ ❡ ♠ó❞✉❧♦✳ ❖ ♠ó❞✉❧♦ é ♦ ❝♦♠♣r✐✲ ♠❡♥t♦ ❞❛ s❡t❛ ❡ ❛ ❞✐r❡çã♦ ❡ ♦ s❡♥t✐❞♦ sã♦ ❛ ❞✐r❡çã♦ ❡ ♦ s❡♥t✐❞♦ ❞❛ s❡t❛ q✉❡ ♦ r❡♣r❡s❡♥t❛✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✸✳
❋✐❣✉r❛ ✸✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❱❡t♦r
●❡r❛❧♠❡♥t❡✱ ✉♠ ♦❜❥❡t♦ ❡♠ q✉❡ s❡ ♣♦❞❡ ❛ss♦❝✐❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ ❞✐r❡çã♦✱ s❡♥t✐❞♦ ❡ ♠ó❞✉❧♦ é ❝❤❛♠❛❞♦ ✉♠ ✈❡t♦r✳ ❆ss✐♠✱ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ é ✉♠ ✈❡t♦r✳ ❯♠ ✈❡t♦r ♥♦ ♣❧❛♥♦ é ✉♠ ♣❛r ♦r❞❡♥❛❞♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s(a, b)✳ ❖s ♥ú♠❡r♦s a ❡ b sã♦ ❝❤❛♠❛❞♦s ❞❡
❝♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r(a, b)✳ P♦r ❡①❡♠♣❧♦✱ v = (6,8)é ✉♠ ✈❡t♦r✳ ❆ ❞✐r❡çã♦ ❞❡st❡ ✈❡t♦r é ❛ ❞✐r❡çã♦ ❞❛ s❡t❛✱ ♦✉ s❡❥❛✱ é ❛ ❞✐r❡çã♦ ❞❛ r❡t❛ ❞❡✜♥✐❞❛ ♣❡❧♦s ♣♦♥t♦sO(0,0) ❡P(6,8)✳ ❖ s❡♥t✐❞♦ ❞❡v é ♦ ❞❡O ♣❛r❛ P ❡ ♦ ♠ó❞✉❧♦ ❞❡v é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ s❡t❛−→OP✱ ♦✉ s❡❥❛✱
é√62+ 82 = 10✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✹✳
■♥✈❡rs❛♠❡♥t❡✱ ✉♠❛ s❡t❛ ♥♦ ♣❧❛♥♦ q✉❡ ♣♦❞❡ s❡r ✐♠❛❣✐♥❛❞❛ ❝♦♠♦ ✉♠❛ ❢♦rç❛ ❞❡ ✐♥t❡♥✲ s✐❞❛❞❡ ✐❣✉❛❧ ❛ ✻ ✉♥✐❞❛❞❡s✱ ❛♣❧✐❝❛❞❛ ❛♦ ♣♦♥t♦ ❖✱ ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ✉♠ ♣❛r ♦r❞❡✲ ♥❛❞♦✳ ◆♦ ❝❛s♦ ❞❛ s❡t❛F ❞❛ ❋✐❣✉r❛ ✺✱ ♦ ♣❛r ♦r❞❡♥❛❞♦ é(6·cos30♦,6·sen30♦) = (3√3,3)✳
❋✐❣✉r❛ ✹✿ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❙❡t❛
❋✐❣✉r❛ ✺✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❋♦rç❛
P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ✉♠ ✈❡t♦r q✉❡ ♥ã♦ ♣❛rt❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❛ ♦r✐❣❡♠✳ ❖❜s❡r✲ ✈❛♥❞♦ ♥❛ ❋✐❣✉r❛ ✻✱ s❡A(x1, y1)❡B(x2, y2)✱ ♦ ✈❡t♦r−→AB = (x2−x1, y2−y1)✳ ❆ s❡t❛ q✉❡
r❡♣r❡s❡♥t❛ ♦ ✈❡t♦r−→AB✱ ♣❛rt✐♥❞♦ ❞❛ ♦r✐❣❡♠✱ ❡ ❛ s❡t❛ ❝♦♠ ♦r✐❣❡♠ ❡♠ A ❡ ❡①tr❡♠✐❞❛❞❡
❡♠ B tê♠ ♦ ♠❡s♠♦ ♠ó❞✉❧♦✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✳
❋✐❣✉r❛ ✻✿ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❱❡t♦r❡s
✷✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s
❙❡❥❛♠ u= (u1, u2)✱ v = (v1, v2)❡ k ∈R✳ ❉❡✜♥✐♠♦s✿
❛✮ ❆ ♦♣❡r❛çã♦ q✉❡ ❛ ❝❛❞❛ ♣❛r ❞❡ ✈❡t♦r❡su❡v ❢❛③ ❝♦rr❡s♣♦♥❞❡r ♦ ✈❡t♦r u+v✱ ❝❤❛♠❛✲s❡
❛❞✐çã♦ ❞❡ ✈❡t♦r❡s✳ ❆ss✐♠✱u+v = (u1+v1, u2+v2).
❜✮ ❆ ❧❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ q✉❡ ❛♦ ♣❛r k ❡ u✱ ♦♥❞❡ k ∈ R ❡ u ✉♠ ✈❡t♦r✱ ❢❛③ ❝♦rr❡s♣♦♥❞❡r
♦ ✈❡t♦rkué ❝❤❛♠❛❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♣♦r ✉♠ ✈❡t♦r✳ ▲♦❣♦✱ku= (ku1, ku2)✳
◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ✈❡t♦r❡s u, v ❡ w✱ t❡♠✲s❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦✿
(A1)u+v =v+u ✭ ▲❡✐ ❝♦♠✉t❛t✐✈❛✮❀
(A2) (u+v) +w=u+ (v +w) ✭▲❡✐ ❛ss♦❝✐❛t✐✈❛✮❀
(A3)u+ 0 =u ✭❊①✐stê♥❝✐❛ ❞❡ ✐❞❡♥t✐❞❛❞❡ ❛❞✐t✐✈❛✮❀
(A4)u+ (−u) = 0 ✭❊①✐stê♥❝✐❛ ❞♦ ♦♣♦st♦✮✳
◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ✈❡t♦r❡s u ❡ v ❡ ♦s ❡s❝❛❧❛r❡s r ❡ s✱ t❡♠✲s❡ ❛s ♣r♦♣r✐❡❞❛❞❡s
❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✿
(M1) (rs)u=r(su)✭▲❡✐ ❛ss♦❝✐❛t✐✈❛✮ ❀
(M2) (r+s)u=ru+su ✭▲❡✐ ❞✐str✐❜✉t✐✈❛✮❀
(M3) r(u+v) = ru+rv ✭▲❡✐ ❞✐str✐❜✉t✐✈❛✮❀
(M4) 1u=u✭❊①✐stê♥❝✐❛ ❞❡ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✮✳
❉❡♠♦♥str❛♠♦s ❛ s❡❣✉✐r ❛ ♣r♦♣r✐❡❞❛❞❡M3✳
❙❡u= (u1, u2) ❡ v = (v1, v2)✱ ❡♥tã♦
r(u+v) =r[(u1, u2) + (v1, v2)] =r(u1+v1, u2+v2)
= (ru1+rv1, ru2+rv2) = (ru1, ru2) + (rv1, rv2)
=r(u1, u2) +r(v1, v2) = ru+rv
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❞❡♠♦♥str❛♠✲s❡ ❛s ❞❡♠❛✐s ♣r♦♣r✐❡❞❛❞❡s✳
❙❡❣✉♥❞♦ ❬✽❪✱ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♥♦ ♣❧❛♥♦ é ♦✉tr❛ s✐t✉❛çã♦ q✉❡ ✐❧✉str❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠❛ s❡t❛ q✉❡ ♥ã♦ ♣❛rt❡ ❞❛ ♦r✐❣❡♠✳ ❘❡♣r❡✲ s❡♥t❛♠♦s ♦ ✈❡t♦r ♣♦s✐çã♦ P ❞❛ ♣❛rtí❝✉❧❛ ♣♦r ✉♠❛ s❡t❛ q✉❡ ♣❛rt❡ ❞❛ ♦r✐❣❡♠ ❡ ♦ ✈❡t♦r
✈❡❧♦❝✐❞❛❞❡ v✱ ♣♦r ✉♠❛ s❡t❛ t❛♥❣❡♥t❡ à tr❛❥❡tór✐❛ ❞❛ ♣❛rtí❝✉❧❛ ❡ ❝♦♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ♥♦
❧✉❣❛r ♦♥❞❡ s❡ ❡♥❝♦♥tr❛ ♥❛q✉❡❧❡ ✐♥st❛♥t❡✳ P❛r❛ r❡♣r❡s❡♥t❛r ❣r❛✜❝❛♠❡♥t❡ ✉♠ ✈❡t♦r ✉s❛✲ s❡ ✉♠❛ s❡t❛✳ ❖ ❧✉❣❛r ♦♥❞❡ ❡st❛ s❡t❛ é ❝♦❧♦❝❛❞❛ ❞❡♣❡♥❞❡ ❞♦ ♣r♦❜❧❡♠❛ q✉❡ ❡stá s❡♥❞♦ ❝♦♥s✐❞❡r❛❞♦✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✼✳
❋✐❣✉r❛ ✼✿ ❘❡♣r❡s❡♥t❛çã♦ ●rá✜❝❛ ❞❡ ✉♠ ❱❡t♦r
❖ ✈❡t♦r (−1)u é ✐♥❞✐❝❛❞♦ ♣♦r −u ❡ ❝❤❛♠❛❞♦ ♦ ♦♣♦st♦ ❞❡ u✳ ■♥❞✐❝❛♠♦s u+ (−v) ♣♦ru−v✳ ◆❛ ❋✐❣✉r❛ ✽✱ ❡stá ✐♥❞✐❝❛❞♦ ✉♠ ✈❡t♦r u ❡ s❡✉ ♦♣♦st♦ −u✳
❖ ✈❡t♦r ku t❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❞❡ u✱ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r s❡t❛s ♣❛r❛❧❡❧❛s✳ ❙❡
k >0✱ ku ❡ u ♣♦ss✉❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✳ ❙❡ k <0✱ ku ❡ u ♣♦ss✉❡♠ ♦s s❡♥t✐❞♦s ❝♦♥✲
❋✐❣✉r❛ ✽✿ ❖ ❖♣♦st♦ ❞❡ ✉♠ ❱❡t♦r
trár✐♦s✳ ❖s ♠ó❞✉❧♦s ❞❡ u ❡ ku ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣♦r ||ku||= |k| · ||u||✱ ♦♥❞❡ ❛ ❜❛rr❛
s✐♠♣❧❡s ✐♥❞✐❝❛ ♦ ♠ó❞✉❧♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❡ ❛ ❜❛rr❛ ❞✉♣❧❛ ✐♥❞✐❝❛ ❛ ♥♦r♠❛ ❞♦ ✈❡t♦r✳
❖❜s❡r✈❡♠♦s q✉❡ ♦ ✈❡t♦r u+v✱ ❡stá ✐♥❞✐❝❛❞♦ ♥❛ ❋✐❣✉r❛ ✾✳ ❆ s❡t❛ q✉❡ r❡♣r❡s❡♥t❛
✉♠❛ ❞❛s ❞✐❛❣♦♥❛✐s ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ é ❛ s♦♠❛ ❡♥tr❡u ❡ v✳
❋✐❣✉r❛ ✾✿ ❆ ❙♦♠❛ ❞❡ ❱❡t♦r❡s
Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡❥❛♠ u ❡v ✈❡t♦r❡s ❡ k ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❊♥tã♦
❛✮ku= 0⇒k = 0 ♦✉u= 0❀
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ u= (u1, u2) ❡k ∈R✳ P♦r ❞❡✜♥✐çã♦✿
ku= (ku1, ku2) = (0,0)
ku1 = 0 ⇒k= 0 ♦✉u1 = 0
ku2 = 0 ⇒k= 0 ♦✉u2 = 0✳ ❆ss✐♠✱
k= 0 ♦✉u= (0,0)
❜✮||u|| ≥0❡ ||u||= 0⇔u= 0❀
❉❡♠♦♥str❛çã♦✳❚♦♠❛♥❞♦u= (u1, u2)✱ t❡♠♦s✿
||u||=pu2
1+u22 ≥0
||u||= 0⇔pu2
1+u22 = 0
u1 =u2 = 0⇔u= 0
❝✮||u+v||6||u||+||v||.
❊stá ❞❡♠♦♥str❛çã♦ s❡rá ❢❡✐t❛ ❛♣ós ✐♥tr♦❞✉③✐r♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❞✉t♦ ❡s❝❛❧❛r✳
✷✳✸ Pr♦❞✉t♦ ❊s❝❛❧❛r ❡ ➶♥❣✉❧♦ ❡♥tr❡ ❱❡t♦r❡s
❉❡✜♥✐çã♦ ✷✳✸ ❖ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞♦s ✈❡t♦r❡su= (u1, u2)❡v = (v1, v2)é ♦ ♥ú♠❡r♦
r❡❛❧ u·v =u1v1+u2v2✳
❉♦✐s ✈❡t♦r❡s sã♦ ortogonais s❡|u+v|=|u−v|✳
❉❡♠♦♥str❛çã♦✳ ❈♦♠♦|u+v|=|u−v|✱ ❡♥tã♦✿
|u+v|2 =|u−v|2
(u+v)(u+v) = (u−v)(u−v)
|u|2+|v|2 + 2uv =|u|2+|v|2 −2uv
4uv = 0⇔uv = 0
❱❡♠♦s q✉❡ ❞♦✐s ✈❡t♦r❡s sã♦ ♦rt♦❣♦♥❛✐s s❡ s❡✉ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❢♦r ③❡r♦✳ ❉♦✐s ✈❡t♦r❡s ❡♠ R2 tê♠ ♣❛♣❡❧ ❡s♣❡❝✐❛❧✳ ❙❡❥❛♠ i = (1,0) ❡ j = (0,1)✳ ❊ss❡s ✈❡t♦r❡s i ❡ j sã♦ ❝❤❛✲
♠❛❞♦s ❞❡ ❜❛s❡ ❝❛♥ô♥✐❝❛✳ ❊❧❡s tê♠ ❝♦♠♣r✐♠❡♥t♦ ✶ ❡ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ❞♦s ❡✐①♦s x ❡ y
♣♦s✐t✐✈♦s✳ ❆ss✐♠✱ q✉❛❧q✉❡r ✈❡t♦r ❡♠R2 ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❡♠ t❡r♠♦s ❞❡i ❡ j✳
❋✐❣✉r❛ ✶✵✿ ❱❡t♦r ♥❛ ❇❛s❡ ❈❛♥ô♥✐❝❛ ❡♠ R2
◗✉❛❧q✉❡r ✈❡t♦r✱w= (x, y) = (x,0)+(0, y)s❡ ❡s❝r❡✈❡w=x(1,0)+y(0,1) = xi+yj✳
Pr♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r
❙❡ u, v ❡ w sã♦ ✈❡t♦r❡s ❡♠ R2 ❡ k é ✉♠ ❡s❝❛❧❛r✱ ❡♥tã♦✿
(1) u·u=||u||2❀
(2) u·v =v·u❀
(3) u·(v+w) = u·v+u·w❀
(4) (ku)·v =k(u·v) =u·(kv)❀ (5) 0·u= 0.
❊ss❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❢❛❝✐❧♠❡♥t❡ ❞❡♠♦♥str❛❞❛s ✉s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r✳ P❛r❛ ❛ ✈❡r✐✜❝❛çã♦ ❞♦s ❞❡t❛❧❤❡s✱ ✐♥❞✐❝❛♠♦s ❬✷❪✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r é✱ t❛❧✈❡③✱ ♦ ❢❛t♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ s♦❜r❡ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❡ ❞♦✐s ✈❡t♦r❡s✳
❚❡♦r❡♠❛ ✷✳✸✳ ❙❡ α ❢♦r ♦ â♥❣✉❧♦ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s ♥ã♦ ♥✉❧♦s u ❡ v✱ ❡♥tã♦
u·v =||u|| · ||v|| ·cosα✳
❉❡♠♦♥str❛çã♦✳ ❆♣❧✐❝❛♥❞♦ ❛ ❧❡✐ ❞♦s ❝♦ss❡♥♦s ❛♦ tr✐â♥❣✉❧♦ ❖❆❇ ❞❛ ❋✐❣✉r❛ ✶✶✱ t❡♠♦s
||−→AB||2 =||−→OA||2+||−−→OB||2−2||−→OA|| · ||−−→OB||cosθ✳ (∗)
❋✐❣✉r❛ ✶✶✿ ➶♥❣✉❧♦ ❡♥tr❡ ❱❡t♦r❡s
▼❛s ||−→OA|| =||u||✱ ||−−→OB||= ||v|| ❡ ||−→AB||= ||u−v||✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ ❡q✉❛çã♦ (∗) s❡ t♦r♥❛
||u−v||2 =||u||2+||v||2−2||u|| · ||v||cosθ✳
❯s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ||u−v||2 =||u||2−2u·v +||v||2✱ t❡♠♦s
−2u·v =−2||u|| · ||v||cosθ✳ ❊♥tã♦✱ u·v =||u|| · ||v||cosθ.
■s♦❧❛♥❞♦cosθ✱ t❡♠♦s✿
cosθ = u·v
||u|| · ||v||
❙❛❜❡✲s❡ q✉❡−1≤cosθ ≤1✱ ❡♥tã♦✿
|cosθ| ≤1
|u·v|
||u|| · ||v|| ≤1
|u·v| ≤ ||u|| · ||v||✳
◗✉❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③
❙❡rá ♣r♦✈❛❞❛ ❛ ♣r♦♣♦s✐çã♦ ✷✳✶✳❝✳
❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r✱ t❡♠♦s✿
||u+v||2 = (u+v)(u+v)
=u·u+u·v+v·u+v·v
||u||2+ 2u·v+||v||2✳
P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ u·v ≤ ||u|| · ||v||✱ ❞❡ ♠♦❞♦ q✉❡✿
||u+v||2 ≤ ||u||2+ 2||u|| · ||v||+||v||2
||u+v||2 ≤(||u||+||v||)2
||u+v|| ≤ ||u||+||v||
✷✳✹ ❆♣❧✐❝❛çõ❡s
✷✳✹✳✶ ❱❡t♦r ❉❡s❧♦❝❛♠❡♥t♦
◗✉❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ ♠♦✈❡✲s❡ ❞❡ ✉♠ ♣♦♥t♦ A(x1, y1) ♣❛r❛ ✉♠ ♣♦♥t♦B(x2, y2)✱ ♦
✈❡t♦r −→AB = (x2−x1, y2 −y1) é ❝❤❛♠❛❞♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛✳ ❆ ❋✐❣✉r❛
✶✷ ✐♥❞✐❝❛ ❛ tr❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞♦ ♣♦♥t♦ A ❛♦ B✳ ❖ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❞❛
♣❛rtí❝✉❧❛ ❡stá ✐♥❞✐❝❛❞♦ ♣❡❧❛ s❡t❛ ❞❡A ♣❛r❛ B✳
❋✐❣✉r❛ ✶✷✿ ❱❡t♦r ❉❡s❧♦❝❛♠❡♥t♦
❙❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♠♦✈❡✲s❡ ❞♦ ♣♦♥t♦A(x1, y1) ♣❛r❛ ♦ ♣♦♥t♦ B(x2, y2) ❡ ❞❡♣♦✐s ♣❛r❛
C(x3, y3)✱ ❡♥tã♦✱ ♦ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ t♦t❛❧ ❞❛ ♣❛rtí❝✉❧❛ é−→AC = (x3−x1, y3−y1)q✉❡
é ❛ s♦♠❛ ❞♦s ❞❡s❧♦❝❛♠❡♥t♦s ♣❛r❝✐❛✐s ❝♦♠♦ ❡stá r❡♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✸✳ ❙❡A, B ❡ C sã♦ três ♣♦♥t♦s q✉❛✐sq✉❡r ❞♦ ♣❧❛♥♦✱ ❡♥tã♦ −→AC =−→AB+−−→BC✳
❋✐❣✉r❛ ✶✸✿ ❱❡t♦r ❉❡s❧♦❝❛♠❡♥t♦ ❚♦t❛❧
❯♠❛ ❛♣❧✐❝❛çã♦ ❞♦ ❛ss✉♥t♦ ❢♦✐ t✐r❛❞❛ ❞❡ ❬✻❪ ❝♦♠ ❛❧❣✉♠❛s ❛❞❛♣t❛çõ❡s✳
❊①❡♠♣❧♦✳ ❯♠ ❛✈✐ã♦ ♣♦❞❡ ✈♦❛r ❛ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❡s❝❛❧❛r ♥♦ ❛r ❞❡ 500km/h✳ ❙❡
♦ ✈❡♥t♦ s♦♣r❛r ♣❛r❛ ♦ ▲❡st❡ ❛ 80km/h✱ q✉❡ ♦r✐❡♥t❛çã♦ ♦ ❛✈✐ã♦ ❞❡✈❡rá s❡❣✉✐r✱ ♣❛r❛ q✉❡
s❡✉ ❝✉rs♦ s❡❥❛ ❞❡30♦❄ ◗✉❛❧ s❡rá ❛ ✈❡❧♦❝✐❞❛❞❡ ❡s❝❛❧❛r ❞♦ ❛✈✐ã♦✱ s❡ ❡❧❡ s❡❣✉✐r ❡ss❡ tr❛❥❡t♦❄
❙♦❧✉çã♦✳ ❖❜s❡r✈❡ à ❋✐❣✉r❛ ✶✹✱ q✉❡ ♠♦str❛ ❛s r❡♣r❡s❡♥t❛çõ❡s ♣♦s✐❝✐♦♥❛✐s ❞♦s ✈❡t♦✲ r❡su ❡ v✱ ❜❡♠ ❝♦♠♦ ❞❡ u−v✳ ❖ ✈❡t♦r u r❡♣r❡s❡♥t❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❛✈✐ã♦ ❡♠ r❡❧❛çã♦
❛♦ s♦❧♦✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ✉♠ ❝✉rs♦ ❞❡ 30♦✳ ❖ â♥❣✉❧♦ ❞❡ ❞✐r❡çã♦ ❞❡ u é ❞❡ 60♦✳ ❖ ✈❡t♦r
v ✐♥❞✐❝❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ✈❡♥t♦✳ ❈♦♠♦v t❡♠ ✉♠ ♠ó❞✉❧♦ ❞❡80❡ ✉♠ â♥❣✉❧♦ ❞❡ ❞✐r❡çã♦ ❞❡ 0♦✱ v = (80,0)✳ ❖ ✈❡t♦r u−v r❡♣r❡s❡♥t❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❛✈✐ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ❛r✳
❆ss✐♠✱ ||u−v||= 500✳ ❙❡❥❛ θ ♦ â♥❣✉❧♦ ❞❡ ❞✐r❡çã♦ u−v
❯t✐❧✐③❛♥❞♦ ❛ ❧❡✐ ❞♦s s❡♥♦s ❛ ❡ss❡ tr✐â♥❣✉❧♦✱ ♦❜t❡♠♦s
sen(φ)
80 ❂
sen(60♦) 500
sen(φ) = 0,138564
φ= 7,966♦
❋✐❣✉r❛ ✶✹✿ ❱❡❧♦❝✐❞❛❞❡ ❊s❝❛❧❛r ❞❡ ✉♠ ❆✈✐ã♦
❊♥tã♦✱
θ = 67,966♦✳
❆♣❧✐❝❛♥❞♦ ❛ ❧❡✐ ❞♦s s❡♥♦s ♥♦✈❛♠❡♥t❡✱ t❡♠♦s
||u||
sen(180♦ −θ)❂
500
sen60♦
||u||= 535,1814✳
❆ ❜úss♦❧❛ ❞♦ ❛✈✐ã♦ ❞❡✈❡r✐❛ ✐♥❞✐❝❛r ✉♠ r✉♠♦ ❞❡ 90♦ −θ✱ ♦✉ s❡❥❛✱ 22,034♦✳ ❊ s❡ ♦ ❛✈✐ã♦ s❡❣✉✐r ❡ss❡ ❝✉rs♦✱ s✉❛ ✈❡❧♦❝✐❞❛❞❡ ❡s❝❛❧❛r ❞❡ s♦❧♦ s❡rá ❞❡✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ 535km/h✳
✷✳✹✳✷ ❱❡t♦r ❘❡s✉❧t❛♥t❡
◆❛ ❋✐❣✉r❛ ✶✺ ❡stã♦ r❡♣r❡s❡♥t❛❞❛s ❞✉❛s ❢♦rç❛sF1 ❡F2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡6N ❡4N✱
❛t✉❛♥❞♦ ❡♠ ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❜❛rr❛✳ ❆ r❡s✉❧t❛♥t❡ ❞❡F1 ❡ F2 é ❛ ❢♦rç❛ F = F1+F2✳
P❛r❛ ❝❛❧❝✉❧❛r F✱ ❡s❝♦❧❤❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ❞❡❝♦♠♣♦♠♦s F1 ❡ F2✳ ❆s
❝♦♠♣♦♥❡♥t❡s ❞❡F1 sã♦x1 = 3
√
3❡ y1 = 3 ❡ ❛s ❞❡ F2 sã♦x2 =−2❡ y2 = 2
√
3✳ ❆ss✐♠✱
F = (3√3−2,2√3 + 3)✳ ❈❛❧❝✉❧❛♥❞♦ ♦ ♠ó❞✉❧♦ ❞❡ F✱ ❡♥❝♦♥tr❛♠♦s ||F|| = 2√13✳ ❆ t❛♥❣❡♥t❡ ❞♦ â♥❣✉❧♦ q✉❡F ❢❛③ ❝♦♠ ♦ ❡✐①♦ xé 24 + 13
√
3
23 ✳
❋✐❣✉r❛ ✶✺✿ ❋♦rç❛ ❘❡s✉❧t❛♥t❡
✷✳✺ Pr♦❥❡çã♦ ❞❡ ❱❡t♦r❡s
❙❡❥❛♠ u = (u1, u2) ❡ v = (v1, v2) ✈❡t♦r❡s ♥ã♦ ♥✉❧♦s ❡ P ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞♦
♣♦♥t♦(u1, u2) s♦❜r❡ ❛ r❡t❛ ❞❡✜♥✐❞❛ ♣♦r(0,0)❡ (v1, v2)✱ ❝♦♥❢♦r♠❡ ♥❛ ❋✐❣✉r❛ ✶✻✳
❋✐❣✉r❛ ✶✻✿ Pr♦❥❡çã♦ ❞❡ ❱❡t♦r❡s
❙❡❥❛θ ♦ â♥❣✉❧♦ ❡♥tr❡ ♦s ✈❡t♦r❡s u❡ v✱ ❝♦♠♦ ||−→OP||=||u||cosθ✱ t❡♠♦s
−→
OP =||u||cosθ v
||v||
♦✉
−→
OP =||u|| u.v
||u||||v|| v
||v|| = u.v v.vv✳
❖❜s❡r✈❡ q✉❡ s❡ θ ≥ 90♦✱ ❛ ❢ór♠✉❧❛ ❝♦♥t✐♥✉❛ ✈á❧✐❞❛✳ ❊st❡ ✈❡t♦r é ❝❤❛♠❛❞♦ ❞❡ Pr♦❥❡çã♦ ❞❡ u s♦❜r❡ v ❡ é ✐♥❞✐❝❛❞♦ ♣♦r Pu
v✳ ❆♦ ♣r♦❥❡t❛r u = (2,3) s♦❜r❡ v = (3,−1)✱
t❡♠♦s
Pu v =
(2,3).(3,−1)
(3,−1).(3,−1)(3,−1) = ( 3
10)(3,−1)✳ ❊①❡♠♣❧♦✳
❛✮❱❡r✐✜q✉❡ q✉❡ ♦ tr✐â♥❣✉❧♦ ❝✉❥♦s ✈ért✐❝❡s sã♦ A(1,1)✱ B = (3,4) ❡ C = (4,−1) é r❡tâ♥❣✉❧♦ ❡♠ ❆✳
❜✮❈❛❧❝✉❧❡ ❛ ♣r♦❥❡çã♦ ❞♦ ❝❛t❡t♦ ❆❇ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛ ❇❈✳ ❝✮❉❡t❡r♠✐♥❡ ♦ ♣é ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ r❡❧❛t✐✈♦ ❛♦ ✈ért✐❝❡ ❆✳
❋✐❣✉r❛ ✶✼✿ ❖ ➶♥❣✉❧♦ ❊♥tr❡ ❱❡t♦r❡s
❙♦❧✉çã♦✳
❛✮ ❇❛st❛ ✈❡r✐✜❝❛r q✉❡ −→AB.−→AC = 0✳ ❈♦♥s✐❞❡r❛♥❞♦ −→AB = u ❡ −→AC = w✱ u = (2,3) ❡
w= (3,−2)✱ ❡♥tã♦ u.w = 2.3 + 3.(−2) = 0✳ ❜✮ ❱❛♠♦s ❝❛❧❝✉❧❛rPu
v ❡✱ ❞❡♣♦✐s✱ ♦ s❡✉ ♠ó❞✉❧♦✳ ❙❡♥❞♦
−→
BA = (−2,−3)❡−−→BC = (1,−5)✱
Pu v =
(−2,−3).(1,−5)
(1,−5).(1,−5) .(1,−5) = 1
2.(1,−5)✳ ❊♥tã♦✱ ❛ ♣r♦❥❡çã♦ ❞❡ u ❡♠ v é
||Pu v||=
1 2.
√
26✳
❝✮ ❙❡❥❛ P(x, y) ♦ ♣é ❞❛ ❛❧t✉r❛ ❛♦ ✈ért✐❝❡A✳ ❊♥tã♦✱
−−→
BP =Pu
v ♦✉(x−3, y−4) = 1
2.(1,−5)✳ ❉♦♥❞❡ P = ( 7 2,
3 2)
✷✳✻ ❊q✉❛çã♦ ❞❛ ❘❡t❛
❈♦♠♦ ❡♠ ❬✷❪✱ ♦s ✈❡t♦r❡s sã♦ ✉♠ ✐♥str✉♠❡♥t♦ ✈❛❧✐♦s♦ ❡♠ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❱❛♠♦s ✐❧✉str❛r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❧✐❣❛❞♦s à ❧✐♥❤❛ r❡t❛✳ ❆ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ r❡t❛ t❡♠ ❛ ❢♦r♠❛
ax+by+c= 0✳
❙❡❥❛P0 = (x0, y0)✉♠ ♣♦♥t♦ ✜①❛❞♦ ♥❛ r❡t❛ ❡ P = (x, y) s❡✉ ♣♦♥t♦ ❣❡♥ér✐❝♦✳ ❊♥tã♦✱
ax0 +by0+c= 0✳
❙✉❜tr❛✐♥❞♦ ❡st❛ ❡q✉❛çã♦ ❞❛ ❛♥t❡r✐♦r✱ ♦❜t❡♠♦s
a(x−x0) +b(y−y0) = 0✳
❊♠ t❡r♠♦s ❞❡ ✈❡t♦r❡s✱ s❡❥❛ v = (a, b)✳ ❈♦♠♦ P −P0 = (x−x0, y −y0)✱ ❛ ú❧t✐♠❛
❡q✉❛çã♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛
(x−x0, y−y0).(a, b) = 0 ♦✉(P −P0).v = 0✳
❋✐❣✉r❛ ✶✽✿ P❡r♣❡♥❞✐❝✉❧❛r✐❞❛❞❡
●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❡st❛ ❡q✉❛çã♦ tr❛❞✉③ ❛ ❝♦♥❞✐çã♦ ❞❡ q✉❡ ♦s ✈❡t♦r❡sP −P0 ❡ v sã♦
♦rt♦❣♦♥❛✐s✳ ❱❡❥❛ ♥❛ ❋✐❣✉r❛ ✶✽✳
❆ss✐♠✱ t♦❞❛ r❡t❛ ♥❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ❣❡r❛❧ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ✈❡t♦r(a, b)✳ ❖ ✈❡t♦r ✉♥✐tár✐♦ ♥❡ss❛ ❞✐r❡çã♦✱u= √ 1
a2+b2(a, b) é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ à r❡t❛✳ ❖s ✈❡t♦r❡s
(b,−a)❡ (−b, a) sã♦ ♥♦r♠❛✐s ❛♦ ✈❡t♦r (a, b)✳ P♦rt❛♥t♦✱ t♦❞❛ r❡t❛ ♥❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ❣❡r❛❧ é ♣❛r❛❧❡❧❛ ❛♦s ✈❡t♦r❡s(b,−a)❡ (−b, a)
❈♦♠♦ ❛♣❧✐❝❛çã♦ ❞❛ ♥♦çã♦ ❞❡ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ ❞✐stâ♥❝✐❛ d ❞❡
✉♠❛ r❡t❛ ❞❡ ❡q✉❛çã♦ax1+by1+c= 0 à ✉♠ ♣♦♥t♦ P0 q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ à r❡t❛✳
❉❡s✐❣♥❛♥❞♦ ♣♦rP ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ❞❛ r❡t❛✱ é ❢á❝✐❧ ✈❡r q✉❡ ❛ ❞✐stâ♥❝✐❛ ♣r♦❝✉r❛❞❛
é ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ♣r♦❥❡çã♦ ❞♦ ✈❡t♦r−−→P P0 ♥❛ ❞✐r❡çã♦ ❞♦ ✈❡t♦r ✉♥✐tár✐♦✱
u= √ 1
a2+b2 ·(a, b)✳
❋✐❣✉r❛ ✶✾✿ ❉✐stâ♥❝✐❛ ❞❡ ✉♠ P♦♥t♦ P0 à ✉♠❛ ❘❡t❛
P♦rt❛♥t♦✱
d=|(−−→P P0).u|=|(x0−x, y0 −y).
(a, b)
√
a2+b2|=
|a(x0−x) +b(y0−y)|
√
a2+b2 .
❈♦♠♦ a.x+b.y+c= 0✱ ♦❜t❡♠♦s
d = |a.x√0+b.y0+c|
a2+b2 ✳
✸ ❙❡çõ❡s ❈ô♥✐❝❛s
❆s s❡çõ❡s ❝ô♥✐❝❛s tê♠ ❡ss❡ ♥♦♠❡ ♣♦r t❡r❡♠ s✐❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❞❡✜♥✐❞❛s ❝♦♠♦ ✐♥t❡r✲ s❡çõ❡s ❞❡ ✉♠ ♣❧❛♥♦ ❝♦♠ ✉♠ ❝♦♥❡✳ ❋❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞❛s ❝✉r✈❛s ❡❧✐♣s❡✱ ♣❛rá❜♦❧❛ ❡ ❤✐♣ér❜♦❧❡✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✐♥❞✐❝❛♠♦s ❬✷❪✱ ❬✻❪ ❡ ❬✽❪✳
✸✳✶ ❊❧✐♣s❡
❉❡✜♥✐♠♦s ❛ ❡❧✐♣s❡ ❝♦♠♦ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ ❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❛ ❞♦✐s ♣♦♥t♦s ✜①♦sF− ❡F é ❝♦♥st❛♥t❡✳ ❯♠❛ ❡❧✐♣s❡ ♣♦❞❡ s❡r ❞❡s❡♥❤❛❞❛ ❝♦♠
♦ ❛✉①í❧✐♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❝♦♠ ❛s ♣♦♥t❛s ❡♠❡♥❞❛❞❛s ♣❛ss❛♥❞♦ ♣♦r ❞♦✐s ♣♦♥t♦s ✜①❛❞♦s ❡♠
F− ❡ F ❡ ✉♠ ❧á♣✐s ♠❛♥t❡♥❞♦ ❛ ❧✐♥❤❛ ❡st✐❝❛❞❛ ❡ ♠♦✈❡♥❞♦✲s❡ ♦ ❧á♣✐s ♥♦ ♣❛♣❡❧✳
❋✐❣✉r❛ ✷✵✿ ❊❧✐♣s❡
❖s ♣♦♥t♦sF− ❡F sã♦ ♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡ ❡ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦F−F é ♦ s❡✉
❝❡♥tr♦✳ P❛r❛ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❝❛rt❡s✐❛♥❛ ❞❡ss❛ ❝✉r✈❛✱ é ♠❛✐s ❢á❝✐❧ ❡s❝♦❧❤❡r ♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ❝♦♠ ♦r✐❣❡♠ ❡♠ s❡✉ ❝❡♥tr♦ ❡ ❡✐①♦ Ox ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ OF✳ ❙❡❥❛♠ −c ❡ c✱
❝♦♠c > 0✱ ❛s ❛❜s❝✐ss❛s ❞❡ F− ❡ F✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ s❡❥❛ 2a ❛ ❝♦♥st❛♥t❡ P F−+P F✱
♦♥❞❡ P = (x, y) é ✉♠ ♣♦♥t♦ ❞❛ ❡❧✐♣s❡✳ P♦♥❞♦ d− =P F− ❡ d=P F ❡✱ ♣❡r❝❡❜❡♥❞♦ q✉❡
d−=
p
(x+c)2+y2 ❡d=p(x−c)2+y2✱ ❛ ❡q✉❛çã♦ ❞❛ ❡❧✐♣s❡ ❛ss✉♠❡ ❛ ❢♦r♠❛
p
(x+c)2+y2+p(x−c)2+y2 = 2a✳
P❛r❛ ❡❧✐♠✐♥❛r ♦s r❛❞✐❝❛✐s✱ ❡❧❡✈❛✲s❡ ❛♦ q✉❛❞r❛❞♦✱ s✐♠♣❧✐✜❝❛✲s❡✱ ❡❧❡✈❛✲s❡ ❛♦ q✉❛❞r❛❞♦ ♥♦✈❛♠❡♥t❡✱ s✉❜st✐t✉✐√a2−c2 ♣♦r ❜ ❡ ❞✐✈✐❞✐♥❞♦ ♦s ♠❡♠❜r♦s ♣♦ra2b2✱ ♦❜t❡♠♦s
x2
a2 +
y2
b2 = 1✳
❆ r❡❝í♣r♦❝❛ t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✱ ♣♦✐s✱ ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ♥ã♦ ♣♦❞❡ s❡r ♥❡❣❛t✐✈❛ ❡✱ s✉♣♦♥❞♦ q✉❡✱ 2a2 − (d−)2+ (d2)
2 = −dd− ♥♦ ✜♥❛❧ t❡rí❛♠♦s q✉❡ a < c✱ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ❆ ❡q✉❛çã♦ ❛❝✐♠❛ é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❡❧✐♣s❡✳
◆♦t❡ q✉❡ ❛ ❡❧✐♣s❡✱ ❝♦♠ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠✱ é s✐♠étr✐❝❛ ❡♠ r❡❧❛çã♦ ❛♦s ❡✐①♦sOx❡ Oy✳
❈❛s♦ tr♦q✉❡♠♦sx♣♦r−x♦✉y ♣♦r−y ❛ ❡q✉❛çã♦ ♥ã♦ s❡ ❛❧t❡r❛✳ ❖s ♣♦♥t♦sA= (±a,0) ❡ B = (0,±b) sã♦ ♦s ✈ért✐❝❡s ❡ ♦s s❡❣♠❡♥t♦s A−A ❡ B−B sã♦ ♦s ❡✐①♦s ❞❛ ❡❧✐♣s❡✳ ❆
❡①❝❡♥tr✐❝✐❞❛❞❡ ❞❛ ❡❧✐♣s❡ é ❞❡✜♥✐❞❛ ♣♦r e = c
a✱ q✉❡ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ s❡ e = 0✱ ✉♠
s❡❣♠❡♥t♦ s❡ e = 1 ❡✱ q✉❛♥❞♦ ❡stá ❡♥tr❡ ✵ ❡ ✶✱ t❡♠♦s ✉♠❛ ❡❧✐♣s❡ q✉❡ é ♠❛✐s ❛❝❤❛t❛❞❛ q✉❛♥❞♦ ❢♦r ♠❛✐s ♣ró①✐♠♦ ❞❡ ✶✳
❱❛♠♦s ♠♦str❛r q✉❡ ❛ r❡t❛ t❛♥❣❡♥t❡ à ✉♠❛ ❡❧✐♣s❡ ❞❛❞❛ ♣❡❧❛ s✉❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❡♠ ✉♠ ❞♦s s❡✉s ♣♦♥t♦sP0 = (x0, y0)✱ t❡♠ ❡q✉❛çã♦
x0x
a2 +
y0y
b2 = 1✳
❉❡r✐✈❛♥❞♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❛ ❡q✉❛çã♦ ❞❛ ❡❧✐♣s❡ ❡ ✐s♦❧❛♥❞♦y′✱ t❡♠♦s✿
y′
=−b
2x
a2y✳
❊♥tã♦✱ ♦ ❞❡❝❧✐✈❡ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ P0 é mt=−
b2x 0
a2y
0❀ ♣♦rt❛♥t♦✱ ❛ ❡q✉❛çã♦
❞❡ss❛ r❡t❛ é
y−y0 =−
b2x 0
a2y 0
(x−x0)
q✉❡ s✐♠♣❧✐✜❝❛♥❞♦ ❡ ❞✐✈✐❞✐♥❞♦ ❡ss❛ ❡q✉❛çã♦ ♣♦r a2b2✱ ❡♥❝♦♥tr❛♠♦s ♦ ❞❡s❡❥❛❞♦✳
Pr♦✈❛♠♦s ❛ s❡❣✉✐r q✉❡ ❛ r❡t❛ ♥♦r♠❛❧ à ❡❧✐♣s❡ ♥✉♠ ❞❡ s❡✉s ♣♦♥t♦s P = (x, y) é ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦F−P F✳ ❙❡❥❛♠m+, mt❡m− ♦s ❞❡❝❧✐✈❡s ❞❛ r❡t❛F P✱ ❞❛ r❡t❛ ♥♦r♠❛❧ QP ❡ ❞❛ r❡t❛ F−P✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ γ ❡ δ ♦s â♥❣✉❧♦s ❞❡ F P ❡QP ❝♦♠ ♦ ❡✐①♦ Ox✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ s♦rt❡ q✉❡✿
❋✐❣✉r❛ ✷✶✿ ❋♦♥t❡ ▲✉♠✐♥♦s❛
m+=tgγ ❡ mt=tgδ✳
P❡❧❛ tr✐❣♦♥♦♠❡tr✐❛✱
tg(γ−δ) = tgγ−tgδ
1 +tgγtgδ✳
❈♦♠♦ r=γ−δ✱ ❧♦❣♦
tg(r) = m+−mt
1 +m+mt✳
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱
tg(i) = mt−m−
1 +mtm−
❉❡✈❡♠♦s ♣r♦✈❛r q✉❡ i=r✱ ♦✉ s❡❥❛✱ tg(i) =tg(r)✱ ♦✉ ❛✐♥❞❛✱
(m++m−)(m2t −1) + 2(1−m−m+)mt = 0
❖❜s❡r✈❛♥❞♦ ❛ ❋✐❣✉r❛ ✷✸✱ ♥♦t❛✲s❡ q✉❡✿
m+=
y
x−c ❡ m−=
y x+c✳
