❛ ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
❆◆❚❖◆■❖ ❊❱❊❘❚❖◆ ❙❖❯❙❆ ❉❆ ❙■▲❱❆
❖ ❆❏❯❙❚❊ ❉❊ ❘❊❚❆❙ P❊▲❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ❊ ❙❊❈➬Õ❊❙ ❉■❉➪❚■❈❆❙ ❉❊ ❙❖▲❯➬➹❖ ▲❙◗ P❆❘❆ ❖ ❊◆❙■◆❖ ▼➱❉■❖
❆◆❚❖◆■❖ ❊❱❊❘❚❖◆ ❙❖❯❙❆ ❉❆ ❙■▲❱❆
❖ ❆❏❯❙❚❊ ❉❊ ❘❊❚❆❙ P❊▲❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ❊ ❙❊❈➬Õ❊❙ ❉■❉➪❚■❈❆❙ ❉❊ ❙❖▲❯➬➹❖ ▲❙◗ P❆❘❆ ❖ ❊◆❙■◆❖ ▼➱❉■❖
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮✱ ❞♦ ❉❡✲ ♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tát✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠á✲ t✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛✲ t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ▼❡✳ ❏✉♥✐♦ ▼♦r❡✐r❛ ❞❡ ❆❧❡♥❝❛r
Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará
Biblioteca do Curso de Matemática
S578a Silva, Antonio Everton Sousa da.
O ajuste de retas pelo método dos mínimos quadrados e secções didáticas de solução LSQ para o ensino médio / Antonio Everton Sousa da Silva. – 2015.
65 f.: il.; enc.; 30 cm.
Dissertação (Mestrado) – Universidade Federal do Ceará, Centro de Ciências, Departamento de Matemática, Mestrado Profissional em Matemática em Rede Nacional, Juazeiro do Norte, 2015.
Orientação: Prof. Me. Júnio Moreira de Alencar.
1. Cálculo. 2. Álgebra. 3. Excel (Programa de computador). I. Título.
❆ ❉❡✉s✳
❆●❘❆❉❊❈■▼❊◆❚❖❙
❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛ ❉❡✉s ♣❡❧❛s ❜❡♥çõ❡s ❡ ❣r❛ç❛s ❛❧❝❛♥ç❛❞❛s ♥❛ ✈✐❞❛ ❡ ❞✉r❛♥t❡ ❡ss❡ ♠❡str❛❞♦✳
❆❣r❛❞❡ç♦ ❛ t♦❞❛ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ✐♥❝❡♥t✐✈♦ ♥❛ ✈✐❞❛✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♥❛♠♦r❛❞❛✱ ❉❛♥✐❡❧❡ ❈♦rr❡✐❛ ❙❛♠♣❛✐♦✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ♠♦t✐✈❛çã♦ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ♠❡✉ ♣r♦❢❡ss♦r ♦r✐❡♥t❛❞♦r✱ ▼❡✳ ❏✉♥✐♦ ▼♦r❡✐r❛✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♥tr✐❜✉✐çã♦✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳
❆❣r❛❞❡ç♦ ❛♦ Pr♦❢✳ ●✉st❛✈♦ ◆♦❣✉❡✐r❛✱ ♣❡❧❛ ❝♦♥tr✐❜✉✐çã♦ ♥❛ ✉t✐❧✐③❛çã♦ ❞♦ ❙♦❢t✇❛r❡ ▲❆❚❊❳✳
❆❣r❛❞❡ç♦ ❛ Pr♦❢❛✳ P❛✉❧❛ ❍❡✈❡❧✐♥❡ ♣❡❧❛ ❝♦rr❡çã♦ ♦rt♦❣rá✜❝❛✳
❆❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❛q✉✐ ♥ã♦ ❢♦r❛♠ ❝✐t❛❞♦s✱ ♠❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞❡ ❢♦r♠❛ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ ▼❡str❛❞♦✳
❘❊❙❯▼❖
❘❡♣r❡s❡♥t❛r ❛tr❛✈és ❞❡ ✉♠❛ r❡t❛ ♦s ❞❛❞♦s ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❞❡ ♠♦❞♦ q✉❡ ♣♦ss❛ ❢❛③❡r ✐♥❢❡rê♥❝✐❛s✱ ♣♦ss✐❜✐❧✐t❛✱ ❛♦ ❡①♣❡r✐♠❡♥t❛r✱ ❡❧❛❜♦r❛r ❤✐♣ót❡s❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ✉♠ ❞❡t❡r✲ ♠✐♥❛❞♦ ❢❡♥ô♠❡♥♦✳ ❊st❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ♦ ♠ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ❝♦♠♦ ❝❛♠✐♥❤♦ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ r❡t❛ q✉❡ ♠❛✐s s❡ ❛❥✉st❛ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳ P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ✐♥✈❡st✐❣❛çã♦✱ ♦♣t♦✉✲s❡ ♣♦r ❢❛③❡r ✉♠❛ ♣❡sq✉✐s❛ ✐♥✲ t❡✐r❛♠❡♥t❡ ❜✐❜❧✐♦❣rá✜❝❛✱ ♣❡r♠✐t✐♥❞♦ r❡❛❧✐③❛r ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ t❡ór✐❝♦ ❛❝❡r❝❛ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ ❞❡♠♦♥str❛çõ❡s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧✱ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ♥❛ ❘❡t❛ ❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❝♦♠♦ ❛✐♥❞❛✱ ❞❡✜♥✐r ♦ ♠ét♦❞♦ ❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s ❡ ❛♣❧✐❝❛r ❛ s♦❧✉çã♦ ▲❙◗ ♥❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ✐♠♣♦ssí✈❡❧ ♥♦ s❡♥t✐❞♦ ✉s✉❛❧✳ ❚❛♠❜é♠ s❡ ❡①♣❧♦r♦✉ ♥❡st❡ ❡st✉❞♦ ♦ ✉s♦ ❙♦❢t✇❛r❡ ❊①❝❡❧ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞❡ ❛♣♦✐♦ ❞✐❞át✐❝♦ ❡ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❋✐♥❛❧✐③❛✲s❡ ❝♦♠ s✉❣❡stã♦ ❞❡ ❛✉❧❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r s♦❜r❡ ♦ t❡♠❛✱ q✉❡ ♣♦r s✉❛ ✈❡③✱ ❡stá ❡♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❛❝❡ssí✈❡❧ ❡ ❝❧❛r❛ ❛♦ ♣r♦❢❡ss♦r✳ ❊ss❛ ❛✉❧❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ❡t❛♣❛s ❞✐❞át✐❝❛s✱ ♥❛ q✉❛❧ ❛ ú❧t✐♠❛✱ ✉t✐❧✐③❛✲s❡ ♦ ❙♦❢t✇❛r❡ ❊①❝❡❧ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❛✉①✐❧✐❛r✳ ◆❛ ♣❡rs♣❡❝t✐✈❛ ❞❡ ✉♠❛ ✜①❛çã♦ ♠❛✐s ♣r♦❢✉♥❞❛ ❞♦ ❝♦♥t❡ú❞♦✱ ❡♥✉♠❡r❛♠♦s ❛♦ ✜♥❛❧ ❞❛ ❛✉❧❛✱ ✈ár✐♦s ❡①❡r❝í❝✐♦s ❞❡ ❛♣❧✐❝❛çã♦✳
❆❇❙❚❘❆❈❚
❚r② t♦ r❡♣r❡s❡♥t ❛ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ❞❛t❛ ♦❢ ❛♥ ❡①♣❡r✐♠❡♥t✱ s♦ t❤❛t ✐t ❝❛♥ ❞r❛✇ ✐♥❢❡r❡♥❝❡s✱ ❡♥❛❜❧❡s t❤❡ ❡①♣❡r✐❡♥❝❡ ❞❡✈❡❧♦♣ ✐♠♣♦rt❛♥t ❤②♣♦t❤❡s✐s ❛❜♦✉t ❛ ♣❤❡♥♦♠❡♥♦♥✳ ❚❤✐s ✇♦r❦ ❛✐♠s t♦ ♣r❡s❡♥t t❤❡ ♠❡t❤♦❞ ♦❢ ❧❡❛st sq✉❛r❡s ❛s ❛ ✇❛② t♦ ✜♥❞ t❤❡ ❧✐♥❡ t❤❛t ❜❡st ✜ts ❛ s❡t ♦❢ ♣♦✐♥ts✳ ❋♦r ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤✐s r❡s❡❛r❝❤✱ ✐t ✇❛s ❞❡❝✐❞❡❞ t♦ ❞♦ ❛ ✇❤♦❧❡ ❧✐t❡r❛t✉r❡✱ ❛❧❧♦✇✐♥❣ t♦ ♠❛❦❡ t❤❡♦r❡t✐❝❛❧ s✉r✈❡② ❛❜♦✉t s♦♠❡ ❝♦♥❝❡♣ts ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❞❡♠♦♥str❛t✐♦♥s✱ ❛♥❛❧②t✐❝❛❧ ❣❡♦♠❡tr② ♦♥ t❤❡ str❛✐❣❤t ❛♥❞ ❧✐♥❡❛r ❛❧❣❡❜r❛✱ ❛s ②❡t✱ ❞❡✜♥❡ t❤❡ ♠❡t❤♦❞ ♦❢ ❧❡❛st sq✉❛r❡s ❛♥❞ ❛♣♣❧② ▲❙◗ s♦❧✉t✐♦♥ t♦ r❡s♦❧✈❡ ❛♥ ✐♠♣♦ss✐❜❧❡ s②st❡♠ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ❆❧s♦ ❡①♣❧♦r❡❞ ✐♥ t❤✐s st✉❞② ✉s✐♥❣ ❊①❝❡❧ s♦❢t✇❛r❡ ❛s ❛ t❡❛❝❤✐♥❣ s✉♣♣♦rt t♦♦❧ ❛♥❞ ✐ts ✉s❡ ✐♥ t❤❡ ❝❧❛ssr♦♦♠✳ ❲❡ ✜♥✐s❤ ✇✐t❤ ❛ ❤✐♥t ♦❢ ❛♥ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r② ❧❡❝t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t✱ ✇❤✐❝❤ ✐♥ t✉r♥✱ ✐s ✐♥ ❛♥ ❛❝❝❡ss✐❜❧❡ ❛♥❞ ❝❧❡❛r ❧❛♥❣✉❛❣❡ t♦ t❤❡ t❡❛❝❤❡r✳ ❚❤✐s ❝❧❛ss ❤❛s ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ❢♦r t❡❛❝❤✐♥❣ st❡♣s✱ ✐♥ ✇❤✐❝❤ t❤❡ ❧❛tt❡r ✐s ♣❡r❢♦r♠❡❞ ✉s✐♥❣ ❊①❝❡❧ s♦❢t✇❛r❡ ❛s ❛♥ ❛✉①✐❧✐❛r② t♦♦❧✳ ❋♦r ❛ ❞❡❡♣❡r ❛tt❛❝❤♠❡♥t ❝♦♥t❡♥t✱ ✇❡ ❡♥✉♠❡r❛t❡❞ t❤❡ ❡♥❞ ♦❢ ❝❧❛ss s❡✈❡r❛❧ ♣r❛❝t✐❝❛❧ ❡①❡r❝✐s❡s✳
❙✉♠ár✐♦
✶ ■◆❚❘❖❉❯➬➹❖ ✾
✷ P❘❊▲■▼■◆❆❘❊❙ ✶✶
✷✳✶ ❊①♣❧♦r❛♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ ❊①♣❧♦r❛♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳✶ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸ ❱❊❚❖❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✶ ❱❡t♦r❡s ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✸✳✷ ❱❡t♦r❡s ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✹ ❊①♣❧♦r❛♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✷✾
✸✳✶ Pr✐♥❝✐♣✐♦ ❞♦ ♠ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷ ❙♦❧✉çã♦ ▲❙◗ ❞❡ ✉♠ s✐st❡♠❛ ▲✐♥❡❛r ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s ❡ ♥ ❡q✉❛çõ❡s ✳ ✳ ✳ ✸✶ ✸✳✸ ❆ r❡t❛ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹ ❚❊❈◆❖▲❖●■❆ ◆❖ ❊◆❙■◆❖ ❉❊ ▼❆❚❊▼➪❚■❈❆ ✹✵
✹✳✶ ❙✉❣❡stã♦ ❞❡ ❛✉❧❛✿ ♠ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ❡ ♦ ❙♦❢t✇❛r❡ ❊①❝❡❧ ✳ ✳ ✳ ✹✸ ✹✳✷ ❊t❛♣❛s ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✶ ✶❛ ❊t❛♣❛✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✷ ✷❛ ❊t❛♣❛✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✸ ✸❛ ❊t❛♣❛✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✹ ✹❛ ❊t❛♣❛✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✷✳✺ ✺❛ ❊t❛♣❛✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
❙❯▼➪❘■❖ ✶✵
■ ✕ P♦r ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ■■ ✕ ❆♣❧✐❝❛çã♦ ❞✐r❡t❛ ❞❛ ❢ór♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✷✳✻ ✻❛ ❊t❛♣❛✿ ❯t✐❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ❊①❝❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ❆t✐✈✐❞❛❞❡✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ Pr♦❝❡❞✐♠❡♥t♦✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✸ ▲✐st❛ ❞❡ ❡①❡r❝í❝✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✺ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ✻✶
❈❛♣ít✉❧♦ ✶
■◆❚❘❖❉❯➬➹❖
❉✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❞❡ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✱ ❝♦♠♦ t❛♠❜é♠ ❞✉r❛♥t❡ ❛ r♦t✐♥❛ ❞❛s ❛t✐✈✐❞❛❞❡s ♣r♦✜ss✐♦♥❛✐s✱ ♣❡r❝❡❜❡✲s❡ q✉❡ ❤á ❛❧❣✉♥s ♠ét♦❞♦s r❡s♦❧✉t✐✈♦s ❞❡ ♣r♦❜❧❡♠❛s ♠❛✲ t❡♠át✐❝♦s q✉❡ ❞❡✐①❛♠ ❝❡rt❛ ✐♥q✉✐❡t❛çã♦✱ ♣♦r s❡r❡♠ ❞❡s❝♦♥❤❡❝✐❞♦s ♦✉ ❛✐♥❞❛ ♣❡❧♦ s❡✉ ❣r❛✉ ❞❡ ❞✐✜❝✉❧❞❛❞❡✳ ❈♦♠♦ é ♦ ❝❛s♦ ❞❡ s✐t✉❛çõ❡s ❡♠ q✉❡ é ♥❡❝❡ssár✐♦ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ♣❛r❛ ❞❡s❝r❡✈❡r ✉♠ ❢❡♥ô♠❡♥♦ ❡s♣❡❝í✜❝♦✳
P❛r❛ t❛♥t♦ ❞❡❝✐❞✐✉✲s❡ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ♣❡sq✉✐s❛ s♦❜r❡ ♦ ♠ét♦❞♦ ❞♦s ✏▼í♥✐♠♦s ◗✉❛❞r❛❞♦s✑ ❡ s✉❛ ❛♣❧✐❝❛çã♦ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❡q✉❛çã♦ ❞❛ r❡t❛✱ ❛♣r❡s❡♥t❛♥❞♦ ✐♥✐❝✐❛❧✲ ♠❡♥t❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s ♣❛r❛ ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦✱ ❝♦♠♦ t❛♠❜é♠ ❡①♣❧♦r❛ ❛♣❧✐❝❛çõ❡s ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❡✱ ❛✐♥❞❛ s✉❣❡r❡ ✉♠❛ ❛✉❧❛ ♣✐❧♦t♦ s♦❜r❡ ♦ ❛ss✉♥t♦ ♣❛r❛ t✉r♠❛s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❆ ♣r♦❜❧❡♠át✐❝❛ q✉❡ ♥♦rt❡✐❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❛ ✐♥✈❡st✐❣❛çã♦ é s✉st❡♥t❛❞❛ ♣❡❧❛ ❥✉st✐✜❝❛t✐✈❛ ❛♥t❡r✐♦r✳ ▼♦t✐✈❛❞♦s ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ ❡st✉❞♦ ❝♦❧❛❜♦r❛t✐✈♦ ♥❛ ár❡❛ ❞❛ ▼❛t❡✲ ♠át✐❝❛ ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❧❡✈❛♥t❛♠♦s ♦ s❡❣✉✐♥t❡ q✉❡st✐♦♥❛♠❡♥t♦✿ ❈♦♠♦ ❛❜♦r❞❛r ❛ r❡t❛ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛tr❛✈és ❞❛s s❡çõ❡s ❞✐❞át✐❝❛s s♦❜r❡ ❛ s♦❧✉çã♦ ▲❙◗✶❄
✶▲❙◗✿ ❙✐❣❧❛ ✐♥❣❧❡s❛ q✉❡ s✐❣♥✐✜❝❛✿ ▲❡❛st ❙q✉❛r❡s ◗✉❛❞r❛t✐❝✳ ❚r❛❞✉çã♦✿ ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ◗✉❛❞rá✲
t✐❝❛s✳
❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✶✵
▼♦t✐✈❛❞♦s ❛ ❡♥❝♦♥tr❛r r❡s♣♦st❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ♣❡♥s❛❞♦✱ tr❛ç♦✉✲s❡ ♦s s❡✲ ❣✉✐♥t❡s ♦❜❥❡t✐✈♦s✿
❆♣r❡s❡♥t❛r ♦ ♠ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s✱ ❡ s❡çõ❡s ❞✐❞át✐❝❛s s♦❜r❡ ❛ s♦❧✉✲ çã♦ ▲❙◗✱ ♥♦ ❛❥✉st❡ ❞❡ r❡t❛s ♣❛r❛ s❡r❡♠ tr❛❜❛❧❤❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
P❛r❛ ❛t✐♥❣✐r ♦ ♣r✐♥❝✐♣❛❧ ♠♦t✐✈♦ ❞❡st❡ ❡st✉❞♦ ❞❡ ♣❡sq✉✐s❛✱ tr❛ç❛r❛♠✲s❡ ❛❧❣✉♥s ♦❜❥❡t✐✈♦s ❡s♣❡❝í✜❝♦s✿
✯ ❊①♣❧♦r❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠ét♦❞♦ ✏▼í♥✐♠♦s ◗✉❛❞r❛❞♦s✑❀
✯ ❊①❡♠♣❧✐✜❝❛r ♣♦r ♠❡✐♦ ❞❡ ❛♣❧✐❝❛çõ❡s ✉♠ ♠ét♦❞♦ ♠❛t❡♠át✐❝♦✿ ❙♦❧✉çã♦ ▲❙◗❀
✯ ❱❡r✐✜❝❛r ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ ✉s♦ ❞❡ t❡❝♥♦❧♦❣✐❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞❡ ❛♣♦✐♦✳
❊ss❛ ♣❡sq✉✐s❛ s❡ ✐♥✐❝✐❛ ❝♦♠ ♦ ❡st✉❞♦ ❞❡ ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s✱ q✉❡ ♣❡r♠✐t✐rã♦ ♦ ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦ ♥❡❝❡ssár✐♦ ❛✜♠ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❛ ❢ór♠✉❧❛✱ q✉❡ ❞❡✜♥❡ ♦s ♣❛râ♠❡tr♦s ❞❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ❞♦s ♠í♥✐♠♦s q✉❛❞r❛❞♦s✳
❊♠ s❡❣✉✐❞❛✱ s❡rá ❡①♣❧♦r❛❞♦ ♦ ♣r✐♥❝í♣✐♦ ♠❛t❡♠át✐❝♦ ❞♦ ♠ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ❡ s✉❛ ♣r✐♥❝✐♣❛❧ ❛♣❧✐❝❛çã♦✱ ♦✉ s❡❥❛✱ ♦ ❛❥✉st❡ ❞❡ ❝✉r✈❛s✳ ◆❡st❡ ♣♦♥t♦ s❡rá ♠♦s✲ tr❛❞♦✱ q✉❡ ❛ ❝✉r✈❛ ❡s❝♦❧❤✐❞❛ ❞❡♣❡♥❞❡rá ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ❞❛❞♦s ♥♦ ❞✐❛❣r❛♠❛ ❞❡ ❞✐s♣❡rsã♦✱ ❡ q✉❡ ❡ss❛ ❝✉r✈❛ ♣♦ss✐❜✐❧✐t❛rá ❢❛③❡r ✐♥❢❡rê♥❝✐❛s ❢✉t✉r❛s s♦❜r❡ ♦ ❡①♣❡r✐♠❡♥t♦ ❛♥❛❧✐s❛❞♦✳
❈❛♣ít✉❧♦ ✷
P❘❊▲■▼■◆❆❘❊❙
◆❡st❡ ❝❛♣ít✉❧♦ ❡①♣õ❡✲s❡ ✐♥✐❝✐❛❧♠❡♥t❡ ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s q✉❡ ❞❛rã♦ s✉♣♦rt❡ t❡ór✐❝♦ ♣❛r❛ ♦ tr❛❜❛❧❤♦✳ ❉❡♣♦✐s ❞❡✜♥❡✲s❡ s♦❧✉çã♦ ▲❙◗ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✱ ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s✱ ♥♦ Rn✱ ❡♥❝♦♥tr❛♥❞♦ ❡ss❛ s♦❧✉çã♦ ✈✐❛ á❧❣❡❜r❛ ❧✐♥❡❛r ❡ ✈✐❛ ❝á❧❝✉❧♦
❞✐❢❡r❡♥❝✐❛❧ ❡ ✜♥❛❧✐③❛✲s❡ ❝♦♠ ❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❡♥✈♦❧✈❡♥❞♦ ❡ss❛ s♦❧✉çã♦ ▲❙◗✳
✷✳✶ ❊①♣❧♦r❛♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛
❉❡✜♥✐çã♦ ✷✳✶✳✶ ✭❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✮✳ ❉❛❞♦s ❞♦✐s ♣♦♥t♦sA(x1, x2) ❡ B(y1, y2)✱
❝❛❧❝✉❧❡♠♦s ❛ ❞✐stâ♥❝✐❛ ❞✭❆✱ ❇✮ ❡♥tr❡ ❡❧❡s ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ três ❝❛s♦s✿
❋✐❣✉r❛ ✷✳✶✿ ❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✷
❚❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡✿
d(A, C) =|x2−x1|❡d(B, C) =|y2−y1|
❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♦ tr✐â♥❣✉❧♦ABC✱ t❡♠♦s✿
[d(A, B)]2
= [d(A, C)]2
+ [d(B, C)]2
[d(A, B)]2
= (x2−x1)2 + (y2−y1)2
d(A, B) = q
(x2 −x1) 2
+ (y2 −y1) 2
❚❡♦r❡♠❛ ✷✳✶✳✶✳ ❉❛❞❛f :R→R ❞❡✜♥✐❞❛ ♣♦r f(x) =ax+b✱ ♦ ❣rá✜❝♦ ❞❡ f é ✉♠❛ r❡t❛✳
❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♣r♦✈❛r q✉❡ três ♣♦♥t♦s q✉❛✐sq✉❡r ❞♦ ❣rá✜❝♦ sã♦ ❝♦❧✐♥❡❛r❡s ❡✱ ♣♦rt❛♥t♦ ❞❡t❡r♠✐♥❛♠ ✉♠❛ r❡t❛✳ ❈♦♥s✐❞❡r❡
P1 = (x1, ax1+b)
P2 = (x2, ax2+b)
P3 = (x3, ax3+b)
P❛r❛ ♣r♦✈❛r♠♦s q✉❡ ❡st❡s três ♣♦♥t♦s sã♦ ❝♦❧✐♥❡❛r❡s é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ ❛ ♠❛✐♦r ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ❞❡❧❡s s❡❥❛ ✐❣✉❛❧ à s♦♠❛ ❞♦s ♦✉tr♦s ❞♦✐s✳ ❙✉♣♦♥❤❛ q✉❡ x1 < x2 < x3✳ ❆ss✐♠✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s t❡♠♦s✿
❞✐stâ♥❝✐❛ ❞❡P1 ❛P2
d(P1, P2) = p
(x2−x1)2+ [(ax2+b)−(ax1+b)]2
=p(x2−x1)2+ [a(x2−x1)]2
d(P1, P2) = (x2−x1) √
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✸
❞✐stâ♥❝✐❛ ❞❡P2 ❛P3
d(P2, P3) = p
(x3−x2)2+ [(ax3+b)−(ax2+b)]2
=p(x3−x2)2+ [a(x3−x2)]2
d(P2, P3) = (x3−x2) √
1 +a2,
❞✐stâ♥❝✐❛ ❞❡P1 ❛P3
d(P1, P3) = p
(x3−x1)2+ [(ax3+b)−(ax1+b)]2
=p(x3−x1)2+ [a(x3−x1)]2
d(P1, P3) = (x3−x1) √
1 +a2.
❉❛í s❡❣✉❡ q✉❡
(x2−x1) √
1 +a2+ (x
3−x2) √
1 +a2 = (x
3−x1) √
1 +a2
❙❡❣✉❡ ❡♥tã♦ q✉❡
d(P1, P3) =d(P1, P2) +d(P2, P3)
▲♦❣♦ ♦s três ♣♦♥t♦s ❡stã♦ ❛❧✐♥❤❛❞♦s ❡✱ ♣♦rt❛♥t♦ ❞❡t❡r♠✐♥❛♠ ✉♠❛ r❡t❛✳
❚❡♦r❡♠❛ ✷✳✶✳✷✳ ❚♦❞❛ r❡t❛ ♥ã♦ ✈❡rt✐❝❛❧ é ♦ ❣rá✜❝♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛y=ax+b✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s A = (x1, y1) ❡ B = (x2, y2) ♥❡ss❛
r❡t❛✳ P♦r ❤✐♣♦t❡s❡ ❛ r❡t❛ é ♥ã♦ ✈❡rt✐❝❛❧✱ ❧♦❣♦ x1 6= x2✳ ◗✉❡r❡♠♦s ❞❡t❡r♠✐♥❛r a ❡ b ❞❡
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✹
❝♦♠ a ❡b s❡♥❞♦ ❛s ✐♥❝ó❣♥✐t❛s✿
ax1+b=y1
ax2+b=y2
=⇒a(x2−x1) = y2−y1 =⇒a=
y2−y1
x2−x1
❙✉❜st✐t✉✐♥❞♦a ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ t❡♠♦s✱
y2−y1
x2−x1
x1 +b =y1 =⇒b=
x2y1−x1y2
x2−x1
❆ss✐♠ ♣r♦✈♦✉✲s❡ q✉❡ ❞❛❞♦s A = (x1, y1) ❡ B = (x2, y2) ❛r❜✐tr❛r✐❛♠❡♥t❡ ❝♦♠
x1 6= x2 ❡①✐st❡ ✉♠❛✱ ❡ s♦♠❡♥t❡ ✉♠❛✱ ❡q✉❛çã♦ y = ax+b✱ t❛❧ q✉❡ y1 = ax1 +b ❡ y2 =
ax2+b✳
❖❜s❡r✈❛çã♦✿ ❊♠ ❛❧❣✉♠❛s ❧✐t❡r❛t✉r❛s✱ ❛ ❡q✉❛çã♦ y = ax +b é t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❞❛ r❡t❛✳
❉❡✜♥✐çã♦ ✷✳✶✳✷ ✭❊q✉❛çã♦ ♣❛r❛♠étr✐❝❛ ❞❛ r❡t❛✮✳ ❆ ❡q✉❛çã♦ ❞❛ r❡t❛ s❡❥❛ ❡❧❛ r❡❞✉③✐❞❛ ♦✉ ❣❡r❛❧✱ r❡❧❛❝✐♦♥❛ ❡♥tr❡ s✐ ❛s ❝♦♦r❞❡♥❛❞❛s (x, y) ❞❡ ✉♠ ♣♦♥t♦ ❣❡♥ér✐❝♦ ❞❛ r❡t❛✳ ❊♥tr❡✲
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✺
❚❡♠♦s q✉❡✱ ♣♦r s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
x−a c−a =
y−b d−b
■❣✉❛❧❛♥❞♦ ❝❛❞❛ ♠❡♠❜r♦ ❛t ❡ ✐s♦❧❛♥❞♦ x❡ y✱ t❡♠♦s✿
x−a
c−a =t=⇒x=a+ (c−a)t y−b
d−b =t=⇒y =b+ (d−b)t
❆s ❡q✉❛çõ❡sx =a+ (c−a)t ❡y =b+ (d−b)t sã♦ ❝❤❛♠❛❞❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛ r❡t❛✳
❆s ❡q✉❛çõ❡s q✉❡ ❞ã♦ ❛s ❝♦♦r❞❡♥❛❞❛s (x, y) ❞❡ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ r❡t❛ ❡♠
❢✉♥çã♦ ❞❡ ✉♠❛ t❡r❝❡✐r❛ ✈❛r✐á✈❡❧ t✿
x=f1(t)❡ y=f1(t)
sã♦ ❝❤❛♠❛❞❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛s r❡t❛✳
❊①❡♠♣❧♦ ✷✳✶✳ ❈♦♥s✐❞❡r❡ ♦s ♣♦♥t♦sA= (3,5)❡B = (1,3)✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛
❡ ❛ ❡q✉❛çã♦ ♣❛r❛♠étr✐❝❛ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s ♣♦♥t♦s✳ ❘❡s♦❧✉çã♦✳
■✮ ❉❡t❡r♠✐♥❛♥❞♦ ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛✿ a = 5−3
3−1 = 1
❙✉❜st✐t✉✐♥❞♦ a ❡ t♦♠❛♥❞♦ ♦ ♣♦♥t♦ A= (3,5)✱ t❡♠♦s 5 = 1.3 +b=⇒b= 2
❆ss✐♠ ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r A ❡ B é y=x+ 2✳
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✻
x= 2 + (1−3)t=⇒x= 2−2t
y = 5 + (3−5)t=⇒y= 5−2t
✷✳✷ ❊①♣❧♦r❛♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r
❉❡✜♥✐çã♦ ✷✳✷✳✶ ✭❊q✉❛çõ❡s ▲✐♥❡❛r❡s✮✳ ❯♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❞❡ n ✈❛r✐á✈❡✐s ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦
a1x1+a2x2+· · ·+anxn=bn
❡♠ q✉❡ x1, x2,· · · , xn sã♦ ✈❛r✐á✈❡✐s✱ a1, a2,· · · , an sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❡q✉❛çã♦ ❡ b ❛ ❝♦♥st❛♥t❡ ❞❛ ❡q✉❛çã♦✳ ❆❧❣✉♥s ❛✉t♦r❡s t❛♠❜é♠ ❝❤❛♠❛♠ b ❞❡ t❡r♠♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡q✉❛çã♦✳
❖❜s❡r✈❛çã♦✿ ●❡r❛❧♠❡♥t❡ ❡♠ ❡q✉❛çõ❡s ❝♦♠ ❛té três ✈❛r✐á✈❡✐s é ❝♦♠✉♠ ✉t✐❧✐③❛r♠♦s ❛s ❧❡tr❛sx✱ y ❡ z ♣❛r❛ r❡♣r❡s❡♥t❛r♠♦s ❡ss❛s ✈❛r✐á✈❡✐s✳
❊①❡♠♣❧♦ ✷✳✷✳
✐✮ 2x1+ 3x2−5x3 = 5
✐✐✮ 2x1−x2−x3 = 0
✐✐✐✮ 0x−0y−0z = 0
❆❜❛✐①♦ t❡♠♦s ❡①❡♠♣❧♦s ❞❡ ❡q✉❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s✿
✐✮ 2x+xy= 5
✐✐✮ 5x2+ 4x= 8
✐✐✐✮ 2√x−8 = 9
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✼
s❡q✉ê♥❝✐❛ ♦✉ ê♥✉♣❧❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s(k1, k2, k3,· · · , kn) é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ s❡
a1k1+a2k2+· · ·+ankn=b
❢♦r ✉♠❛ s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✳
❊①❡♠♣❧♦ ✷✳✸✳ ❉❛❞❛ ❛ ❡q✉❛çã♦ ❧✐♥❡❛r2x1+ 3x2−x3+ 4x4 = 3✱ ✈❡r✐✜q✉❡ s❡ ❛s s❡q✉ê♥❝✐❛s
❛❜❛✐①♦ sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦✳
✐✮ ✭✶✱ ✹✱ ✸✱ ✕ ✷✮
❘❡s♦❧✉çã♦✳ ❆ s❡q✉ê♥❝✐❛ ❛❜❛✐①♦ é s♦❧✉çã♦✱ ♣♦✐s 2.(1) + 3.(4)−(3) + 4.(−2) = 3 é
✉♠❛ s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✳
✐✐✮ (2,−1,3,4)
❘❡s♦❧✉çã♦✳ ❆ s❡q✉ê♥❝✐❛ ❛❝✐♠❛ ♥ã♦ é s♦❧✉çã♦✱ ♣♦✐s 2.(2) + 3.(−1)−(3) + 4.(4) = 3
♥ã♦ é ✉♠❛ s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✳
❉❡✜♥✐çã♦ ✷✳✷✳✸ ✭❙✐st❡♠❛ ▲✐♥❡❛r✮✳ ❉❡♥♦♠✐♥❛♠♦s ❞❡ s✐st❡♠❛ ❧✐♥❡❛r ❝♦♠m ❡q✉❛çõ❡s ❡n ✐♥❝ó❣♥✐t❛s ✉♠ ❝♦♥❥✉♥t♦ ❞❡m (m≥1)❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ ♥❛s ✐♥❝ó❣♥✐t❛s x1, x2, x3,· · · , xn ♦ s✐st❡♠❛ q✉❡ ❡stá r❡♣r❡s❡♥t❛❞♦ ❛❜❛✐①♦✿
S :
a11x1 + a12x2 + a13x3 + · · · + a1nxn = b1
a21x1 + a22x2 + a23x3 + · · · + a2nxn = b2
a31x1 + a32x2 + a33x3 + · · · + a3nxn = b3
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
am1x1 + am2x2 + am3x3 + · · · + amnxn = bm
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✽ S:
a11k1 + a12k2 + a13k3 + · · · + a1nkn = b1
a21k1 + a22k2 + a23k3 + · · · + a2nkn = b2
a31k1 + a32k2 + a33k3 + · · · + a3nkn = b3
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
am1k1 + am2k2 + am3k3 + · · · + amnkn = bm
❊①❡♠♣❧♦ ✷✳✹✳ ❱❡r✐✜q✉❡ s❡ ♦s ❝♦♥❥✉♥t♦s ❛❜❛✐①♦ sã♦ s♦❧✉çã♦ ❞♦ s✐st❡♠❛
S :
2x+ 4y+z = 10
x+y+z =−1
3x−y+z =−3
✐✮ (1,2,−4)é s♦❧✉çã♦✱ ♣♦✐s
2·(1) + 4·(3)−4 = 10 ✭s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✮ 1 + 2−4 =−1 ✭s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✮ 3·(1)−2−4 =−3 ✭s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✮
✐✮ (−5,5,0)♥ã♦ é s♦❧✉çã♦✱ ♣♦✐s
2·(−5) + 4·(5) + 0 = 10 ✭s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛✮ −5 + 5 + 0 =−1 ✭s❡♥t❡♥ç❛ ❢❛❧s❛✮ 3·(−5)−5 + 0 =−3 ✭s❡♥t❡♥ç❛ ❢❛❧s❛✮
❆ss✐♠✱ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❞❡m ❡q✉❛çõ❡s ❡n✐♥❝ó❣♥✐t❛s é ❞✐t♦ ♣♦ssí✈❡❧ ✭❝♦♠♣❛✲ tí✈❡❧✮ ❡ ❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦❀ P♦ssí✈❡❧ ❡ ✐♥❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s❀ ■♠♣♦ssí✈❡❧ ✭✐♥❝♦♠♣❛tí✈❡❧✮ q✉❛♥❞♦ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦✳
✷✳✷✳✶ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ▼❛tr✐③❡s
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✶✾
a11 a12 · · · a1n
a21 a22 · · · a2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn
· x1 x2 ✳✳✳ xm = b1 b2 ✳✳✳ bm
♦✉A·X =B ♦♥❞❡
A=
a11 a12 · · · a1n
a21 a22 · · · a2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn
♠❛tr✐③ ❞♦s ❝♦❡✜❝✐❡♥t❡s✱
X = x1 x2 ✳✳✳ xm
♠❛tr✐③ ❞❛s ✐♥❝ó❣♥✐t❛s ❡
B = b1 b2 ✳✳✳ bm
❛ ♠❛tr✐③ ❞♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳
❍á t❛♠❜é♠ ♦✉tr❛ ♠❛♥❡✐r❛ q✉❡ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ♦ s✐st❡♠❛ ❛ ✉♠❛ ♠❛tr✐③✱ ❝♦♠♦
✈❡♠♦s ❛❜❛✐①♦✿
a11 a12 · · · a1n b1
a21 a22 · · · a2n b2
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn bm
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✵
❊①❡♠♣❧♦ ✷✳✺✳ ❉❛❞♦ ♦ s✐st❡♠❛ ❛❜❛✐①♦ ❡s❝r❡✈❛✲♦ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✳
x+ 4y+z = 10
x+y+z = 4
3x−y+z = 2
❘❡s♦❧✉çã♦✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ▼❛tr✐③❡s
▼❛tr✐③ ❢♦r♠❛❞❛ ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s✿
1 4 1
1 1 1
3 −1 1
▼❛tr✐③ ❢♦r♠❛❞❛ ♣❡❧❛s ✐♥❝ó❣♥✐t❛s✿
x y z
▼❛tr✐③ ❢♦r♠❛❞❛ ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s✿
10 4 2 ▲♦❣♦✱ t❡r❡♠♦s✿
1 4 1
1 1 1
3 −1 1 · x y z 10 4 2
▼❛tr✐③ ❛♠♣❧✐❛❞❛ ❞♦ s✐st❡♠❛
1 4 1 10
1 1 1 4
3 −1 1 2
✷✳✸ ❱❊❚❖❘
P❛r❛ ♥♦ss♦ ❡st✉❞♦ ❞❡ ✈❡t♦r ❝♦♥s✐❞❡r❛r❡♠♦sRn ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✳
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✶
r❡♣r❡s❡♥t❛ ❛ ✐✲és✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ✈❡t♦r✳
❙❡♥❞♦ ♦s ❡❧❡♠❡♥t♦s ❞❡Rn✈❡t♦r❡s✱ ❛ s♦♠❛ ❞❡ ❞♦✐s ✈❡t♦r❡s−→v = (x1, x2,· · · , xn) ❡−→w = (y1, y2,· · · , yn) é ❞❛❞❛ ♣♦r
−
→v +→−w = (x1, x2,· · · , xn) + (y1, y2,· · · , yn) = (x1+y1, x2+y2,· · · , xn+yn)
❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ✈❡t♦r−→v = (x1, x2,· · · , xn)♣❡❧♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ α✱ ❝❤❛♠❛❞♦ ❡s❝❛❧❛r✱ é ❞❡✜♥✐❞❛ ♣♦r
α−→v =α(x1, x2,· · · , xn) = (αx1, αx2,· · · , αxn)
❊①❡♠♣❧♦ ✷✳✻✳ ❙❡♥❞♦ −→v = (1,2,4) ❡ −→w = (2,4,2) ✈❡t♦r❡s ❞♦ R3 ❡ α = 2 ✉♠ ♥ú♠❡r♦
r❡❛❧✱ ❝❛❧❝✉❧❡✿
❛✮ −→v +−→w
❜✮ α−→w ❘❡s♦❧✉çã♦✳
❛✮ −→v +−→w = (1 + 2,2 + 4,4 + 2) = (3,6,6)
❜✮ α−→w = 2(2,4,2) = (4,8,4)
✷✳✸✳✶ ❱❡t♦r❡s ♥♦
R
2❈♦♥s✐❞❡r❛♥❞♦ ✐♥✐❝✐❛❧♠❡♥t❡✱ q✉❡ R2 r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✷
❉❡✜♥✐çã♦ ✷✳✸✳✶✳ ❙❡❥❛♠A(a1, a2)❡B(b1, b2)♣♦♥t♦s ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❡−→v = (x, y)✉♠
✈❡t♦r ❞❡ R2✳ ❉✐③✲s❡ q✉❡ ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ −→AB r❡♣r❡s❡♥t❛ ♦ ✈❡t♦r −→v s❡
x=b1−a1
y=b2−a2
✷✳✸✳✷ ❱❡t♦r❡s ♥♦
R
3❖ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ♣❛r❛ ♦R3✳ ❈♦♥s✐❞❡r❛♥❞♦ R3 ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s
♣♦♥t♦s ❞♦ ❡s♣❛ç♦✱ ✐st♦ é✱ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞♦ s✐st❡♠❛ ❞❡ ❡✐①♦s ♠✉t✉❛♠❡♥t❡ ♣❡r♣❡♥❞✐✲ ❝✉❧❛r❡s✱ Ox✱ Oy✱ ❡ Oz✱ ❝♦♠ ❛ ♠❡s♠❛ ♦r✐❣❡♠ ❡♠ O✱ ❞❡✜♥✐♠♦s ❝♦♠♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦
−→
P Q✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s q✉❡ ❡stã♦ ♥♦ s❡❣♠❡♥t♦ P Q ❞❡ ❡①tr❡♠♦s P ❡ Q✱ ❝♦♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡♠P ♣♦♥t♦ ✜♥❛❧ ❡♠ Q✳
❉❡✜♥✐çã♦ ✷✳✸✳✷✳ ❙❡❥❛♠ A(a1, a2, a3) ❡ B(b1, b2, b3)♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❡ −→v = (x, y, z)✉♠
✈❡t♦r ❞❡ R3✳ ❉✐③✲s❡ q✉❡ ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ −→AB r❡♣r❡s❡♥t❛ ♦ ✈❡t♦r −→v s❡
x=b1−a1
y=b2−a2
z =b3−a3
❉❡✜♥✐çã♦ ✷✳✸✳✸ ✭❈♦♠❜✐♥❛çã♦ ▲✐♥❡❛r✮✳ ❙❡❥❛♠Rn♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✱−→v 1,→−v 2,· · · ,−→vn ✈❡t♦r❡s ♣❡rt❡♥❝❡♥t❡s ❛Rn✳ ❉✐③✲s❡ q✉❡ ♦ ✈❡t♦r−→v é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡−→v1,−→v2,· · · , −
→vn s❡ ❡①✐st✐r❡♠ ❡s❝❛❧❛r❡sa1, a2,· · · , an ♣❡rt❡♥❝❡♥t❡s ❛ R✱ t❛❧ q✉❡
−
→v =a1−→v1 +a2−→v2 +· · ·+an−→vn
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✸
❉❡✜♥✐çã♦ ✷✳✸✳✹ ✭❉❡♣❡♥❞ê♥❝✐❛ ❡ ■♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r✮✳ ❙❡❥❛♠ Rn ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧
❡−→v 1,−→v 2,· · · ,−→v n✈❡t♦r❡s ♣❡rt❡♥❝❡♥t❡s ❛Rn✳ ❉✐③❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦{−→v 1,−→v 2,· · · ,−→vn} é ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡(LD)✱ ♦✉ q✉❡ ♦s ✈❡t♦r❡s−→v 1,−→v 2,· · ·,−→vn sã♦LD✱ s❡ ❛ ❡q✉❛çã♦
a1−→v1+a2−→v2· · ·+ai−→v i+· · ·+an−→v n= 0
❛❞♠✐t✐r s♦❧✉çã♦ ♥ã♦✲tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱ ❛❧❣✉♠ai 6= 0✳ ❈❛s♦ ❝♦♥trár✐♦✱ ✐st♦ é✱a1 =a2 =· · ·=
an= 0 ♦ ❝♦♥❥✉♥t♦ {−→v1,−→v2,· · · ,−→v n} s❡rá ❝❤❛♠❛❞♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ (LI)✳
❚❡♦r❡♠❛ ✷✳✸✳✶✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s é ▲■ s❡✱ ❡ s♦♠❡♥t❡ s❡ ♥❡♥❤✉♠ ❞❡❧❡s ❢♦r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♦✉tr♦s✳
❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ s❡rá ❢❡✐t❛ ❡♠ ❞✉❛s ♣❛rt❡s✳ ✶❛ P❛rt❡
❙❡❥❛ {−→v 1,−→v 2,· · · ,−→vn} ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛❧❣✉♠ −→v i✱ ❝♦♠ 1 ≤ i≤n✱ ❞❡ss❡ ❝♦♥❥✉♥t♦{−→v 1,−→v 2,· · ·,−→vn}s❡❥❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ❞❡♠❛✐s✱ ❧♦❣♦ ❡①✐st❡♠ ❡s❝❛❧❛r❡sa1,· · · , ai−1, ai+1,· · · , an t❛✐s q✉❡
−
→vi =a1−→v1+a2−→v2+· · ·+ai−1−→vi−1+ai+1−→vi+1+· · ·+an−→v n
r❡❛rr❛♥❥❛♥❞♦ ❡st❛ ❡q✉❛çã♦ t❡♠♦s
a1−→v1+a2−→v2+· · ·+ai−1−→v i−1+ (−1)−→vi+ai+1−→vi+1+· · ·+an−→v n= 0
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✹
❙✉♣♦♥❤❛♠♦s q✉❡−→v1,−→v2,· · · ,−→v n sã♦LD✱ t❡♠♦s ❡♥tã♦
a1−→v 1+a2−→v 2+· · ·+ai−→vi +· · ·+an−→vn = 0, ❝♦♠1≤i≤n❡ai 6= 0 ⇒
⇒ −ai−→v i =a1−→v1 +a2→−v2 +· · ·+ai−1−→vi−1+ai+1−→v i+1+· · ·+an−→v n⇒
⇒ −→vi =−1
ai
(a1−→v 1+a2→−v 2+· · ·+ai−1−→vi−1+ai+1−→v i+1+· · ·+an−→v n)
❆❜s✉r❞♦✱ ♣♦✐s ♥❡♥❤✉♠ −→v i ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ❞❡♠❛✐s✳
❉❡✜♥✐çã♦ ✷✳✸✳✺ ✭Pr♦❞✉t♦ ❡s❝❛❧❛r✮✳ ❉❛❞♦s ♦s ✈❡t♦r❡s −→v = (x1, x2,· · · , xn) ❡ −→u =
(y1, y2,· · · , yn) ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♦✉ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s❡rá ❞❛❞♦ ♣♦r✿
−
→v · −→u = (x1y1+x2y2+· · ·+xnyn)
❙❡❥❛−→v = (x1, x2,· · · , xn)✉♠ ✈❡t♦r ❡♠ Rn✱ t❡♠♦s q✉❡✿
−
→v · −→v ≥0.
❉❡ ❢❛t♦✱
−
→v · −→v = (x1x1+x2x2+· · ·+xnxn) = x2 1 +x
2
2+· · ·+x 2
n ≥0
❉❡✜♥✐çã♦ ✷✳✸✳✻ ✭◆♦r♠❛ ❞❡ ✉♠ ✈❡t♦r✮✳ ❙❡♥❞♦−→v = (x1, x2,· · · , xn)❛ ♥♦r♠❛ ♦✉ ❝♦♠♣r✐✲ ♠❡♥t♦ ❞❡−→v✱ r❡♣r❡s❡♥t❛❞♦ ♣♦rk −→v k ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r✿
k −→v k= (−→v · −→v)1/2 = q
x2
1+x22+· · ·+x2n
❉❡✜♥✐çã♦ ✷✳✸✳✼ ✭P❡r♣❡♥❞✐❝✉❧❛r✐❞❛❞❡ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s✮✳ ❉❛❞♦ ✉♠ ✈❡t♦r−→v ❡ ♦✉tr♦ ✈❡t♦r
−
→u s❡ ❡ss❡s ❞♦✐s ✈❡t♦r❡s ❢♦r❡♠ ♦rt♦❣♦♥❛✐s ♦✉ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ❡ss❡s
❞♦✐s ✈❡t♦r❡s s❡rá ③❡r♦✱ ♦✉ s❡❥❛✱
−
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✺
✳
❚❡♦r❡♠❛ ✷✳✸✳✷ ✭❞❡ P✐tá❣♦r❛s ❛tr❛✈és ❞❡ ✈❡t♦r❡s✮✳ ❉❛❞♦sA✱ B ❡ C três ♣♦♥t♦s ❞♦ Rn✱ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ✈❡t♦r❡s−→a =B−C✱−→b =C−A❡−→c =B−A✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦s ✈❡t♦r❡s
−
→b ❡ −→c s❡❥❛♠ ♦rt♦❣♦♥❛✐s✱ ❛ss✐♠−→b
· −→c = 0 ❚❡♠♦s q✉❡ −→a =B−C =B−A−(C−A) = −
→c −−→b✳ ❆ss✐♠
k −→a k2=−→a · −→a = (−→c −−→b)·(−→c −−→b ) =−→c · −→c −2→−b · −→c +−→b ·−→b
▲♦❣♦
k −→a k2=k−→b k2 +k −→c k2
❆ ♣r♦♣♦s✐çã♦ q✉❡ s❡rá ❞❡♠♦♥str❛❞❛ ❛ s❡❣✉✐r é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✳
Pr♦♣♦s✐çã♦ ✷✳✸✳✸✳ ❙❡❥❛ A✱ B ❞♦✐s ♣♦♥t♦s ❞♦ Rn ❡ s❡❥❛ Ω ♦ ❝♦♥❥✉♥t♦✱ ❝♦♥t❡♥❞♦ A✱ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s C ❞❡ Rn t❛✐s q✉❡ C−A s❡❥❛ ♦rt♦❣♦♥❛❧ ❛ B−A✳ ◆❡st❛ ❝♦♥❞✐çõ❡s✱ ♣❛r❛ t♦❞♦ C ❡♠ Ω ✱
kB−Ak≤kB −C k
♦✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ C ❡♠ Ω✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ B ❛ A é ♠❡♥♦r ♦✉ ✐❣✉❛❧ à ❞✐stâ♥❝✐❛ ❞❡ B ❛ C✳
❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❡♠♦s q✉❡ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s t❡♠♦s✱
kB −Ck2=kB−Ak2 +kC−Ak2 .
❈♦♠♦kC−Ak2
≥0✱ r❡s✉❧t❛kB−C k2
≥kB−Ak2 ❡✱ ♣♦rt❛♥t♦✱
kB−Ak≤kB −C k
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✻
✈á❧✐❞♦ q✉❡
|−→v · −→u| ≤k −→v k · k −→u k
❉❡♠♦♥str❛çã♦✿ ❙❡−→u ♦✉−→v é ♦ ✈❡t♦r ♥✉❧♦✱ ❡♥tã♦ ❛ ❞❡♠♦♥str❛çã♦ s❡ r❡❞✉③ ❛ ✈❡r✐✜❝❛r ❛ ✐❣✉❛❧❞❛❞❡✱ ③❡r♦ ✐❣✉❛❧ ❛ ③❡r♦✳
❙✉♣♦♥❤❛−→u ❡ −→v ✈❡t♦r❡s ♥ã♦ ♥✉❧♦s✳ P❛r❛ q✉❛❧q✉❡r t ∈Rt❡♠♦s q✉❡
(t−→u +−→v)·(t−→u +−→v )≥0
❖✉ s❡❥❛✱ ♣❛r❛ q✉❛❧q✉❡r t∈R✱
− →u · −→u t2
+ 2−→u · −→v t+−→v · −→v ≥0 ✭✷✳✶✮
❚♦♠❛♥❞♦ p(t) = −→u · −→u t2+ 2−→u · −→v t+−→v · −→v , t ∈ R✳ P♦r ✷✳✶✱ p é ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦✲ ♠✐❛❧ ♥ã♦ ♥❡❣❛t✐✈❛ ❡ ❝♦♠♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ q✉❛❞rát✐❝♦ é ♥ã♦ ♥❡❣❛t✐✈♦✱ s❡❣✉❡ q✉❡ ♦ ❞✐s❝r✐♠✐♥❛♥t❡∆ ❞❡p(t) é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♣♦s✐t✐✈♦✳ P♦r t❛♥t♦✱
∆ = 4(−→u · −→v)2
−4(−→u · −→u)(−→v · −→v ) = 4(−→u · −→v )2
−4k −→u k2k −→v k2≤0
❙❡❣✉❡|−→v · −→u|2
≤k −→v k2
· k −→u k2✳ P♦rt❛♥t♦
|−→v · −→u| ≤k −→v k · k −→u k✳
✷✳✹ ❊①♣❧♦r❛♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧
❉❡✜♥✐çã♦ ✷✳✹✳✶ ✭❉❡r✐✈❛❞❛s ♣❛r❝✐❛✐s✮✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s✱ x ❡ y✳ ❆ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❞❡ f ❡♠ r❡❧❛çã♦ ❛ x é ❛q✉❡❧❛ ❢✉♥çã♦✱ ❞❡♥♦t❛❞❛ ♣♦r ∂f
∂x✱ t❛❧ q✉❡ s❡✉s ✈❛❧♦r❡s ❢✉♥❝✐♦♥❛✐s ❡♠ q✉❛❧q✉❡r ♣♦♥t♦(x, y✮ ♥♦ ❞♦♠í♥✐♦ ❞❡ f s❡❥❛♠ ❞❛❞♦s ♣♦r
∂f
∂x = lim∆x→0
f(x+ ∆x, y)−f(x, y) ∆x
s❡ ♦ ❧✐♠✐t❡ ❡①✐st✐r✳
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✼
f ❡♠ r❡❧❛çã♦ ❛ y é ❛q✉❡❧❛ ❢✉♥çã♦✱ ❞❡♥♦t❛❞❛ ♣♦r ∂f
∂y✱ t❛❧ q✉❡ s❡✉s ✈❛❧♦r❡s ❢✉♥❝✐♦♥❛✐s ❡♠ q✉❛❧q✉❡r ♣♦♥t♦(x, y) ♥♦ ❞♦♠í♥✐♦ ❞❡ f s❡❥❛♠ ❞❛❞♦s ♣♦r
∂f
∂y = lim∆y→0
f(x, y+ ∆y)−f(x, y) ∆y
s❡ ❡ss❡ ❧✐♠✐t❡ ❡①✐st✐r✳
❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ s❡ ❧✐♠✐t❛ ❛s ❢✉♥çõ❡s ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s✳ ❊♥tr❡t❛♥t♦✱ r❡s❛❧t❛✲ ♠♦s q✉❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❛ ❢✉♥çõ❡s ❞❡n ✈❛r✐á✈❡✐s✱ ♠❡❞✐❛♥t❡ ✉♠❛ ❞❡✜♥✐çã♦ ♠❛✐s ❢♦r♠❛❧✳
◆♦ ❝á❧❝✉❧♦ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s ❝♦♠ ✈❛❧♦r❡s r❡❛✐s✱ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ❡✈✐t❛r ❣r❛♥❞❡s ❝á❧❝✉❧♦s✱ r❡❞✉③✲s❡ ❛ ❞❡r✐✈❛❞❛ ❛♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♥s✐❞❡r❛ ❛ ❢✉♥çã♦ ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❝❛❞❛ ✈❡③✱ ♠❛♥t❡♥❞♦ ✜①❛s ❛s ❞❡♠❛✐s✱ ♦✉ s❡❥❛✱ s❡rã♦ ❝♦♥s✐❞❡r❛❞❛s ❝♦♥st❛♥t❡s✳
❊①❡♠♣❧♦ ✷✳✼✳ ❈❛❧❝✉❧❡ ❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡
f(x, y) = (8xy−y4)2
+ 6y2x
❘❡s♦❧✉çã♦✳ ❈❛❧❝✉❧❡♠♦s ∂f
∂x ❝♦♥s✐❞❡r❛♥❞♦x❝♦♠♦ ✈❛r✐á✈❡❧ ❡y❝♦♠♦ ✉♠❛ ❝♦♥st❛♥t❡✱ ❛ss✐♠ t❡♠♦s✿
∂f
∂x = 2(8xy−y
4)8y+ 6y2
= 16y(8xy−y4) + 6y2
❈❛❧❝✉❧❛♥❞♦ ❛❣♦r❛✱ ∂f
∂y ❝♦♥s✐❞❡r❛♥❞♦y❝♦♠♦ ✈❛r✐á✈❡❧ ❡x❝♦♠♦ ✉♠❛ ❝♦♥st❛♥t❡✱ ❛ss✐♠ t❡♠♦s✿ ∂f
∂y = 2(8xy−y
4
)(8x−4y3) + 12yx
❉❡✜♥✐çã♦ ✷✳✹✳✷✳ ❙❡❥❛f(x, y)✉♠❛ ❢✉♥çã♦ ❛ ✈❛❧♦r❡s r❡❛✐s ❡ s❡❥❛(x0, y0)A ∈Df✳ ❉✐③❡♠♦s q✉❡(x0, y0)é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞❡f ❡♠ A s❡✱ ♣❛r❛ t♦❞♦ (x, y)❡♠ A✱
f(x0, y0)≤f(x, y)
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ✷✽
q✉❡(x0, y0)é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❞❡f ❡♠ A s❡✱ ♣❛r❛ t♦❞♦ (x, y) ❡♠ A✱
f(x0, y0)≥f(x, y)
❉❡✜♥✐çã♦ ✷✳✹✳✸✳ ❙❡❥❛ f(x, y) ❞❡ ❝❧❛ss❡C2✳ ❆ ❢✉♥çã♦ H ❞❛❞❛ ♣♦r
H(x, y) =
∂2f
∂x2(x, y)
∂2f
∂x∂y(x, y) ∂2f
∂x∂y(x, y) ∂2f ∂y2(x, y)
❉❡♥♦♠✐♥❛✲s❡ ❤❡ss✐❛♥♦ ❞❡f✳ ❖❜s❡r✈❡ q✉❡
H(x, y) = ∂ 2f
∂x2(x, y)
∂2f
∂y2(x, y)−
∂2f
∂x∂y(x, y)
2
❚❡♦r❡♠❛ ✷✳✹✳✶✳ ❙❡❥❛f(x, y✮ ❞❡ ❝❧❛ss❡C2 ❡
(x0, y0)um ponto interior de Df✳ ❙✉♣♦♥❤❛✲ ♠♦s q✉❡ (x0, y0) s❡❥❛ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ f✳ ❊♥tã♦
■✮ ❙❡ ∂2f
∂x2(x0, y0)>0 ❡ H(x0, y0)>0✱ ❡♥tã♦ (x0, y0) s❡rá ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f✳
■■✮ ❙❡ ∂2f
∂x2(x0, y0)<0 ❡ H(x0, y0)>0✱ ❡♥tã♦ (x0, y0) s❡rá ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ f✳
❈❛♣ít✉❧♦ ✸
❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙
◗❯❆❉❘❆❉❖❙
❚❡♥t❛r ♠♦❞❡❧❛r ❛tr❛✈és ❞❡ ✉♠❛ ❝✉r✈❛ ✉♠ ❢❡♥ô♠❡♥♦ ♥❛t✉r❛❧ ♦✉ ♠❡s♠♦ ❡①♣❡r✐✲ ♠❡♥t❛❧✱ ❞❡ ♠♦❞♦ q✉❡ ♣♦ss❛ ❞❡s❝r❡✈ê✲❧♦ ❛tr❛✈és ❞❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ✈❛r✐á✈❡✐s✱ ❛✐♥❞❛ é ✉♠ ❣r❛♥❞❡ ❞❡s❛✜♦ ❡♥❢r❡♥t❛❞♦ ♣❡❧❛ ❝✐ê♥❝✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ♥ã♦ ❤á ✉♠❛ ❞✐str✐✲ ❜✉✐çã♦ r❡❣✉❧❛r ❞❡ s❡✉s ❞❛❞♦s ❡♠ ❛♥á❧✐s❡s✳ ▼❛s✱ ❙✐❧✈❛ ●✳✭✷✵✶✵✱ ♣✳ ✹✮ ❬✾❪ ❡st❛❜❡❧❡❝❡ q✉❡ ✏❖ ❛❥✉st❡ ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ✐st♦ é✱ ❛ ❜✉s❝❛ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❡ ♠❡❧❤♦r ❞❡s❝r❡✈❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✱ é ❢❡rr❛♠❡♥t❛ ✐♠♣r❡s❝✐♥❞í✈❡❧ ♥♦ ♣r♦❝❡ss♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ❞❡ ✉♠ ❢❡♥ô✲ ♠❡♥♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ q✉❛♥❞♦ s❡ ❛❞♠✐t❡ q✉❡ ❡❧❡ s❡❥❛ ❞❡s❝r✐t♦✱ ♠❡s♠♦ q✉❡ ♣❛r❝✐❛❧♠❡♥t❡✱ ♣♦r ❞❡t❡r♠✐♥❛❞♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦✑✳
❙❡ ❞♦ ❢❡♥ô♠❡♥♦ ❡♠ ❡st✉❞♦ ♣✉❞❡r ❞❡❞✉③✐r ✉♠❛ ❝✉r✈❛ q✉❡ ♠❡❧❤♦r s❡ ❛❥✉st❡ ❛♦s ❞❛❞♦s ❝♦❧❤✐❞♦s✱ é ♣r♦✈á✈❡❧ q✉❡ ❡ss❛ ❝✉r✈❛ ♥ã♦ ❛❜r❛♥❥❛ t♦❞♦s ♦s ❞❛❞♦s✱ ♦❝♦rr❡♥❞♦ ✐st♦ ❤❛✈❡rá ✉♠❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❞❛❞♦s ❝♦❧❤✐❞♦s ❡ ♦s ❞❛❞♦s ❞❛ ❝✉r✈❛✳ ❖ ▼ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ❝♦♥s✐st❡ ❡♠ ♠✐♥✐♠✐③❛r ❡ss❛s ❞✐❢❡r❡♥ç❛s ❞❡ ♠♦❞♦ ❛ ❛♣r♦①✐♠❛r ♦ ♠á①✐♠♦ ♣♦ssí✈❡❧ ❡ss❛ ❝✉r✈❛ ❞♦s ♣♦♥t♦s✱ ♦✉ s❡❥❛✱ ❡st❛ té❝♥✐❝❛ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❞❡t❡r♠✐♥❛r ❛ ❝✉r✈❛ q✉❡ ♠❛✐s s❡ ❛❥✉st❛ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✳ ❆❜❛✐①♦ s❡rá ❞❛❞❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ❡ ✐♥t✉✐t✐✈❛ ❞♦ ♠ét♦❞♦✳
❈❆P❮❚❯▲❖ ✸✳ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✸✵
✸✳✶ Pr✐♥❝✐♣✐♦ ❞♦ ♠ét♦❞♦ ❞♦s ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s
◆❛ ✜❣✉r❛ ❛❜❛✐①♦ ❡stã♦ r❡♣r❡s❡♥t❛❞♦s ✉♠❛ ❝✉r✈❛ C ❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s
{(x1, y1), (x2, y2),· · · ,(xn, yn)} ❢♦r❛ ❞❡❧❛✳ ❆♦ ❧♦♥❣♦ ❞❡ss❛ ❝✉r✈❛✱ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡ xi✱ ❤❛✈❡rá ✉♠yi✳ ❍❛✈❡rá ❛✐♥❞❛ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦yi ❡ ♦ ✈❛❧♦r ❝♦rr❡s♣♦♥❞❡♥t❡ ♥❛ ❝✉r✈❛✱ ♦✉ s❡❥❛✱ yi✳ ❚❛❧ ❞✐❢❡r❡♥ç❛ s❡rá ❝❤❛♠❛❞❛ ❞❡ di✳ ❆ss✐♠ ♣❛r❛ ❝❛❞❛ xi✱ t❡♠♦s di = yi −yi✳ ❊ss❛s ❞✐❢❡r❡♥ç❛s ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡ ❞❡s✈✐♦s✱ ❊rr♦s ♦✉ r❡sí❞✉♦s✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠ s❡r ♣♦s✐t✐✈❛s✱ s❡ ♦ ♣♦♥t♦ ❡st✐✈❡r ❛❜❛✐①♦ ❞❛ ❝✉r✈❛✱ ♥❡❣❛t✐✈❛ s❡ ❡st✐✈❡r ❛❝✐♠❛ ❞❛ ❝✉r✈❛ ♦✉ ③❡r♦ s❡ ♦ ♣♦♥t♦ ❡st✐✈❡r s♦❜r❡ ❛ ❝✉r✈❛✳ ❆ss✐♠✱ ♣❛r❛ ❡✈✐t❛r ❞✐st♦rçõ❡s ♥♦s ❝á❧❝✉❧♦s✱ ✉s❛✲s❡ ♦ q✉❛❞r❛❞♦ ❞❡ss❛s ❞✐❢❡r❡♥ç❛s✱ t❛❧✈❡③ ♣♦r ✐ss♦ ♦ ♥♦♠❡ ✏▼í♥✐♠♦s ◗✉❛❞r❛❞♦s✑✳ ❖❜s❡r✈❡♠♦s ♦ ❣rá✜❝♦ ❛ s❡❣✉✐r✳
❯♠❛ ♠❡❞✐❞❛ q✉❡ ♥♦s ❞❛rá ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣r❡❝✐s❛ ❞♦s ♣♦♥t♦s ❛ ❝✉r✈❛ é ♣r♦♣♦r❝✐♦♥❛❞❛ ♣❡❧♦ ✈❛❧♦r D = d2
1+d 2 2+d
2
3 +· · ·+d 2
n✳ ◗✉❛♥t♦ ♠❡♥♦r ❢♦r ❡ss❡ ✈❛❧♦r✱
♠❛✐s ♣ró①✐♠♦ ♦s ♣♦♥t♦ ❡st❛rã♦ ❞❛ ❝✉r✈❛✳ ❆ss✐♠ ♣♦❞❡♠♦s ❞❡✜♥✐r✿
❉❡✜♥✐çã♦ ✸✳✶✳✶✳ ❉❡ t♦❞❛s ❛s ❝✉r✈❛s q✉❡ s❡ ❛❥✉st❡♠ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✱ ❛ q✉❡ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❛♣r❡s❡♥t❛r ♦ ♠❡♥♦r ✈❛❧♦r
D=d2
1+d 2 2+d
2
3+· · ·+d 2
n
❈❆P❮❚❯▲❖ ✸✳ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✸✶
❊ss❛ ❞❡✜♥✐çã♦ ♥♦s ❞✐③ q✉❡✱ s❡ ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ♦s ♣♦♥t♦s ❞❛ ❝✉r✈❛ ❡ ♦s ♣♦♥t♦s ❞♦ ❡①♣❡r✐♠❡♥t♦ ❡♠ q✉❡stã♦ ❢♦r ❛ ♠❡♥♦r ♣♦ssí✈❡❧ ❡♥tã♦ ❡ss❛ ❝✉r✈❛ é ❛ q✉❡ ♠❡❧❤♦r s❡ ❛❥✉st❛ ❛ ❡ss❡ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✳
✸✳✷ ❙♦❧✉çã♦ ▲❙◗ ❞❡ ✉♠ s✐st❡♠❛ ▲✐♥❡❛r ❝♦♠ ❞✉❛s ✐♥✲
❝ó❣♥✐t❛s ❡ ♥ ❡q✉❛çõ❡s
❉❡✜♥✐çã♦ ✸✳✷✳✶ ✭❙♦❧✉çã♦ ▲❙◗✮✳ ❙❡❥❛S ♦ s✐st❡♠❛ ❧✐♥❡❛r ❛❜❛✐①♦ ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s
S :
a11x + a12y = b1
a21x + a22y = b2
✳✳✳ ✳✳✳ ✳✳✳
an1x + an2y = bn
❉✐③❡♠♦s q✉❡(x0, y0)é ✉♠❛ s♦❧✉çã♦LSQ❞❡Ss❡(x, y) = (x0, y0)t♦r♥❛r ♠í♥✐♠❛ ❛ ❞✐stâ♥❝✐❛
❞❡
− →b =
b1 b2 ✳✳✳ bn
❛ −→v =
a11x + a12y
a21x + a22y
✳✳✳ ✳✳✳ an1x + an2y
❚❡♦r❡♠❛ ✸✳✷✳✶✳ ❙❡❥❛ ❙ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❛❜❛✐①♦ ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s
S :
a11x + a12y = b1
a21x + a22y = b2
✳✳✳ ✳✳✳ ✳✳✳
an1x + an2y = bn
♦♥❞❡−→v1 =
a11 a21 ✳✳✳ an1
,−→v2 =
a12 a21 ✳✳✳ an2
❡−→b = b1 b2 ✳✳✳ bn
❈❆P❮❚❯▲❖ ✸✳ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✸✷ ①✐❧✐❛r ❛❜❛✐①♦✿ AS : − →v
1 · −→v1x+→−v1 · −→v2y =
− →
b · −→v1
− →v
1 · −→v2x+→−v2 · −→v2y =
− →
b · −→v2
❉❡♠♦♥str❛çã♦ ✈✐❛ á❧❣❡❜r❛✳ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ✈❡t♦r❡s −→v1 ❡ −→v2 ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥✲ ❞❡♥t❡s✱ ♣♦✐s s❡ ❢♦ss❡♠ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s t❡rí❛♠♦s ✉♠ s✐st❡♠❛ ❞❡n❡q✉❛çõ❡s ❡ ❝♦♠ ✉♠❛ ✈❛r✐á✈❡❧✱ q✉❡ é ✐rr❡❧❡✈❛♥t❡ ♣❛r❛ ♦ tr❛❜❛❧❤♦✳ ◗✉❡r❡♠♦s ❞❡t❡r♠✐♥❛r ✉♠ ✈❡t♦r −→v0 t❛❧
q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡−→v0 ❡ −→b ✱ s❡❥❛ ♠í♥✐♠❛✱ ♦✉ s❡❥❛✱ −→b − −→v0 ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦✿
❚♦♠❛♥❞♦−→v0 ♣❡rt❡♥❝❡♥t❡ ❛♦ ♣❧❛♥♦✱ ❧♦❣♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
− →v 0 =
a11x0 + a12y0
a21x0 + a22y0
✳✳✳ ✳✳✳
an1x0 + an2y0
♦✉ −→v0 =x0→−v1 +y0−→v2
♣❛r❛ ❛❧❣✉♠x0 ❡ y0 r❡❛❧✳
❱❛♠♦s ❡s❝r❡✈❡r ♦ s✐st❡♠❛ ❙ ♥❛ ❢♦r♠❛ ✈❡t♦r✐❛❧✿
S :x−→v1 +y−→v2 =
− →
b
P❛r❛ q✉❡(−→b − −→v0)s❡❥❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s−→v1 ❡−→v2✱ ❞❡✈❡♠♦s t❡r
(−→b − −→v0)⊥ −→v1 ❡(
− →
❈❆P❮❚❯▲❖ ✸✳ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✸✸
❧♦❣♦
(−→b − −→v0)· −→v1 = 0 ❡ (
− →
b − −→v0)· −→v2 = 0
❛ss✐♠ t❡♠♦s
(−→b − −→v0)· −→v1 = 0
(−→b − −→v0)· −→v2 = 0 ⇒
(−→b −x0−→v1 −y0−→v2)· −→v1 = 0
(−→b −x0−→v1 −y0−→v2)· −→v2 = 0 ⇒ ⇒ − →b
· −→v1 −x0−→v1 · −→v1 −y0−→v2 · −→v1 = 0 −
→b
· −→v1 −x0−→v1 · −→v2 −y0−→v2 · −→v2 = 0 ⇒ − →b
· −→v1 −x0−→v1 · −→v1 −y0−→v2 · −→v1 = 0 −
→b
· −→v1 −x0−→v1 · −→v2 −y0−→v2 · −→v2 = 0 ⇒
⇒
x0−→v1 · −→v1 +y0−→v2 · −→v1 =
− →
b · −→v1
x0−→v1 · −→v2 +y0−→v2 · −→v1 =
− →
b · −→v1
❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❡♥❝♦♥tr❛♠♦sx0❡y0❝♦♠♦ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❛✉①✐❧✐❛r ❡ q✉❡ é s♦❧✉çã♦
LSQ❞♦ s✐st❡♠❛ S✳
❚❡♦r❡♠❛ ✸✳✷✳✶✳ ❙❡❥❛ ❙ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❛❜❛✐①♦ ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s
S :
a11x + a12y = b1
a21x + a22y = b2
✳✳✳ ✳✳✳ ✳✳✳
an1x + an2y = bn
♦♥❞❡−→v1 =
a11 a21 ✳✳✳ an1
,−→v2 =
a12 a21 ✳✳✳ an2
❡−→b = b1 b2 ✳✳✳ bn
❙✉❛ s♦❧✉çã♦ ✭♦✉ s♦❧✉çõ❡s✮ LSQ é ❛ s♦❧✉çã♦ ✭♦✉ sã♦ s♦❧✉çõ❡s✮ ❞♦ s✐st❡♠❛ ❛✉✲ ①✐❧✐❛r ❛❜❛✐①♦✿ AS : − →v
1 · −→v1x+→−v1 · −→v2y =
− →
b · −→v1
− →v
1 · −→v2x+→−v2 · −→v2y =
− →
b · −→v2
❉❡♠♦♥str❛çã♦ ✈✐❛ ❈á❧❝✉❧♦✳
❈❆P❮❚❯▲❖ ✸✳ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✸✹
❧✐♥❡❛r ❞❡ −→v1 ❡−→v2✱ ❧♦❣♦ t❡♠♦s✿
−
→v0 =x−→v1 +y−→v2 = (xa11+ya12, xa21+ya22,· · · , xa1n+ya2n)
❚♦♠❛♥❞♦D(x, y) ❝♦♠♦ ♦ q✉❛❞r❛❞♦ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ −→b ❛−→v0✱ t❡♠♦s
k −→v0 −
− →
b k2
=D(x, y) = (xa11+ya12−b1) 2
+· · ·+ (xa1n+ya2n−bn)
2
D(x, y) = n
XXX
i=i
(xa1i+ya2i−bi)
2
P❡❧♦ t❡♦r❡♠❛ ✷✳✹✳✶✱ ❞❡♠♦♥str❛✲s❡ q✉❡ ❛ ❢✉♥çã♦ D(x, y) ♣♦ss✉✐ ♠í♥✐♠♦ ❧♦❝❛❧✱
❧♦❣♦ ❛♣r❡s❡♥t❛ ∂D ∂x = 0 ❡
∂D ∂y = 0✳
❉❡r✐✈❛♥❞♦ D ❡♠ ❢✉♥çã♦ ❞❡ x ❡ ✐❣✉❛❧❛♥❞♦ ❛ 0✱ t❡♠♦s✿
∂D
∂x = 0⇒ ∂D
∂x = n
X
i=1
2(xa1i+ya2i−bi)·a1i ⇒ n
X
i=1
2(xa1i+ya2i−bi)·a1i = 0 ⇒
n
X
i=1
(xa1i+ya2i−bi)·a1i = 0 ⇒ n
X
i=1
❈❆P❮❚❯▲❖ ✸✳ ❖ ▼➱❚❖❉❖ ❉❖❙ ▼❮◆■▼❖❙ ◗❯❆❉❘❆❉❖❙ ✸✺
⇒x n
X
i=1
a1ia1i+y n
X
i=1
a2ia1i− n
X
i=1
bia1i = 0⇒
⇒x−→v1 · −→v1 +y−→v1 · −→v2 −−→b · −→v1 = 0⇒x−→v1 · −→v1 +y−→v1 · −→v2 =−→b · −→v1 = 0
❉❡r✐✈❛♥❞♦ D ❡♠ ❢✉♥çã♦ ❞❡ y✱ t❡♠♦s✿
∂D
∂y = 0⇒ ∂D
∂y = n
X
i=1
2(xa1i+ya2i−bi)·a2i ⇒ n
X
i=1
2(xa1i+ya2i−bi)·a2i = 0 ⇒
n
X
i=1
(xa1i+ya2i−bi)·a2i = 0 ⇒ n
X
i=1
(xa1ia2i+ya2ia2i−bia2i) = 0
⇒x n
X
i=1
a1ia2i+y n
X
i=1
a2ia2i− n
X
i=1
bia2i = 0⇒
⇒x−→v1 · −→v2 +y−→v2 · −→v2 −−→b · −→v2 = 0⇒x−→v1 · −→v2 +y−→v2 · −→v2 =−→b · −→v2 = 0
❆ss✐♠ ✜❝❛♠♦s ❝♦♠ ♦ s✐st❡♠❛ ❛✉①✐❧✐❛r✿
SA:
x−→v1 · −→v1 +y−→v1 · −→v2 =
− →
b · −→v1 = 0
x−→v1 · −→v2 +y−→v2 · −→v2 =
− →
b · −→v2 = 0