Open Aplicações da geometria analítica na resolução de problemas

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❆♣❧✐❝❛çõ❡s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛

♥❛ ❘❡s♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s

♣♦r

❆ss✐❝❧❡r♦ ❈❛✈❛❧❝❛♥t❡ ❚❡♦t♦♥✐♦ ❞❡ ▲❛❝❡r❞❛

s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦ ✴ ✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ♦ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

♥❛ ❘❡s♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s

♣♦r

❆ss✐❝❧❡r♦ ❈❛✈❛❧❝❛♥t❡ ❚❡♦t♦♥✐♦ ❞❡ ▲❛❝❡r❞❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ❲✐❧❜❡r❝❧❛② ●♦♥ç❛❧✈❡s ▼❡❧♦ ✲ ❯❋❙

❆❣r❛❞❡❝✐♠❡♥t♦s

❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛ ❉❡✉s✱ ♣♦r t❡r ♠❡ ♣❡r♠✐t✐❞♦ r❡❛❧✐③❛r ❡ss❡ tr❛❜❛❧❤♦✱ ♣r♦✈❡♥❞♦✲ ♠❡ t♦❞❛s ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ♠❡str❛❞♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ s❡❣✉r❛ ❡ ♣❡❧❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛❞❛s ❛♦ t❡①t♦✳

❆♦ Pr♦❢✳ ❊❞✉❛r❞♦ ❲❛❣♥❡r✱ ♣♦r t❡r ♠❡ ✐♥s♣✐r❛❞♦✱ ❝♦♠ s✉❛s ❛❜♦r❞❛❣❡♥s s♦❜r❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❛ ❞❡s❡♥✈♦❧✈❡r ♦ t❡♠❛✳

➚ ♠✐♥❤❛ ❡s♣♦s❛ ▼❛r✐❛ ❞♦ ❆♠♣❛r♦✱ ♣❡❧♦ ❛♣♦✐♦ ❝♦♥st❛♥t❡ ❞❡s❞❡ ♦ ♠♦♠❡♥t♦ ❞♦ ❡①❛♠❡ ❞❡ ❛❝❡ss♦ ❛té ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ♣❡❧♦ ❝♦♠♣❛rt✐❧❤❛♠❡♥t♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✳ ❆♦s ♠❡♠❜r♦s ❞❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ♣❡❧❛ ✐♥✐❝✐❛t✐✈❛ ❞❡ ❡st✐♠✉❧❛r ❛ ♠❡❧❤♦r✐❛ ❞♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ❛tr❛✈és ❞❡st❡ ♠❡str❛❞♦✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖▼❆❚ ❛ ♥í✈❡❧ ♥❛❝✐♦♥❛❧ ❡ ❛ ♥í✈❡❧ ❧♦❝❛❧✱ ♣❡❧♦ s✉♣♦rt❡ ❞❛❞♦ ❛♦ ♣r♦❣r❛♠❛✳

❆♦ ♠❡✉ ✜❧❤♦ ■❣♦r ♣♦r ❡stá s❡♠♣r❡ ♣r❡s❡♥t❡✳

➚ ❈❆P❊❙✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ ❡ ❝r❡❞✐❜✐❧✐❞❛❞❡ ❞✐s♣❡♥s❛❞❛ ❛♦ P❘❖❋▼❆❚✳

➚ ♠✐♥❤❛ ♠ã❡ ▼❛r✐❛ ■✈❡t❡✳ ❆♦ ♠❡✉ ♣❛✐ ❋r❛♥❝✐s❝♦ ❞❡ ❆ss✐s ✭❡♠ ♠❡♠ór✐❛✮✳

❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ é ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❛❜♦r❞❛❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ♣❛r❛ s❡r ✉s❛❞❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ❣❡r❛❧✳ ❆ ●❡♦♠❡tr✐❛ ❆♥❛✲ ❧ít✐❝❛ t❡♠ s❡✉ ♠❛✐♦r ✈❛❧♦r✱ ❡①❛t❛♠❡♥t❡ ♦♥❞❡ ❡❧❛ é ✐♥❡s♣❡r❛❞❛✱ ♥❛s ❛♣❧✐❝❛çõ❡s ❡♠ ♦✉tr♦s r❛♠♦s ❞❛s ❝✐ê♥❝✐❛ ❡①❛t❛s✳ ❖ ♦❜❥❡t✐✈♦ é ♠♦str❛r ❝♦♠♦ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛✱ ❡ ❝♦♠♦ ♣♦❞❡♠♦s ❢❛③❡r ♣❛r❛ ❡♥❢❛t✐③❛r ♣❛r❛ ♦ ❛❧✉♥♦ q✉❡ ❡❧❛ ♥ã♦ é ✉♠❛ ♣❛rt❡ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ s❡ ❡♥❝❡rr❛ ❡♠ s✐ ♠❡s♠❛✳ ❋❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ✈❡t♦r✐❛❧✱ q✉❛♥❞♦ ❢♦r ❝♦♥✈❡♥✐❡♥t❡✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❛s ❞❡♠♦str❛çõ❡s ❞❡ ❢ór♠✉❧❛s ❡ r❡s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s s❡ t♦r♥❡♠ ♠❛✐s s✐♠♣❧❡s✳ ❆❝r❡❞✐t❛♠♦s q✉❡ ❢❛③❡♥❞♦ ✉♠❛ ❛r✲ t✐❝✉❧❛çã♦ ❡♥tr❡ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❜❡♠ ♥❛t✉r❛❧✱ ❡st❛r❡♠♦s ♠❡❧❤♦r❛♥❞♦ ♦ ♥í✈❡❧ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦s ❛❧✉♥♦s✳

P❛❧❛✈r❛s✕❈❤❛✈❡✿ ❱❡t♦r❡s✱ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❡ ❆♣❧✐❝❛çõ❡s✳

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ✐t ✐s ❛ ♣r♦♣♦s❛❧ ♦❢ ❛♣♣r♦❛❝❤ ♦❢ ❆♥❛❧②t✐❝ ●❡♦♠❡tr② t♦ ❜❡ ✉s❡❞ ❛s ❛ t♦♦❧ ♦♥ r❡s♦❧✉t✐♦♥ ♦❢ ♣r♦❜❧❡♥s ✐♥ ❣❡♥❡r❛❧✳ ❚❤❡ ❆♥❛❧✐t✐❝ ●❡♦♠❡tr② ❤❛s ✐ts ❣r❡❛t❡r ✈❛❧✉❡ ❛①❛❝t② ✇❤❡r❡ ✐t ✐s ✉♥❡s♣❡❝t❡❞✱ ✐♥ ❛♣❧✐❝❛t✐♦♥s ✐♥ ♦t❤❡r ❜r❛♥❝❤❡s ♦❢ ❡①❛❝t ❡❝✐❡♥❝❡✳ ❚❤❡ ♣✉r♣♦s❡ ✐t ✐s t♦ s❤♦✉ ❤♦♠ t❤❡ ❆♥❛❧②t✐❝ ●❡♦♠❡tr② ❝❛♥ ❜❡ ❛♣♣❧✐❡❞✱ ❛♥❞ ❤❛✉ ❡❛ ✇❡ ❞♦ t♦ ❡♠♣❧❛s✐③❡ t♦ st✉❞❡♥ts t❤❛❧ t❤❡ ❆♥❛❧②t✐❝ ●❡♦♠❡tr② ✐s ♥♦t ❛ ♣❛rt ♦❢ ▼❛t❡♠❛t✐❝s ✇❤✐❝❤ ❡♥❞s ✐ts❡❧❢✳ ❲❡ ✇✐❧❧ ♠❛❦ ✈❡❝t♦r ❛♣♣r♦❛❝❤✱ ✇❤❡♥ ✐t ✐s ❝♦♥✈✐♥✐❡♥t ♠❛❦✐♥❣ t❤❡ st❛t♠❡♥ts ✐♥ ❢♦r♠✉❧❛s ❛♥❞ r❡s♦❧✉t✐♦♥s ♦❢ ♣r♦❜❧❡♥s✱ ❜❡❝♦♠❡ ♠♦r❡ s✐♠♣❧❡✳ ❲❡ ❜❡❧✐❛✈❡ t❤❛t ♠❛❦✐♥❣ ❛ ❛rt✐❝✉❧❛t✐♦♥ ❜❡t✇❡❡♥ ❆♥❛❧②t✐❝ ●❡♦♠❡tr② ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥✱ ✐♥ ❛ ✈❡r② ♥❛t✉r❛❧ ✇❛②✱ ✇❡ ❛r❡ ✐♠♣r♦✈✐♥❣✱ t❤❡ ❧❡✈❡❧ ♦❢ st✉❞❡♥ts ❧❡❛r♥✐♥❣✳

❑❡②✇♦r❞s✿ ❱❡❝t♦rs✱ ❆♥❛❧✐t✐❝ ●❡♦♠❡tr②✱ ❆♣❧✐❝❛t✐♦♥s✳

❙✉♠ár✐♦

✶ ❱❡t♦r❡s ❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✶

✶✳✶ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❊q✉✐♣♦❧ê♥❝✐❛ ❞❡ ❙❡❣♠❡♥t♦s ❖r✐❡♥t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ ❱❡t♦r❡s ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✻ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✻✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ❆❞✐çã♦ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✻✳✷ Pr♦♣r✐❡❞❛❞❡s ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❱❡t♦r❡s ♣♦r ❊s❝❛❧❛r❡s ✳ ✳ ✳ ✳ ✾ ✶✳✼ Pr♦❞✉t♦ ■♥t❡r♥♦ ❡♠ ❚❡r♠♦s ❞❡ ❈♦♦r❞❡♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✼✳✶ ◆♦r♠❛ ❞❡ ✉♠ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✼✳✷ ❱❡t♦r❡s P❡r♣❡♥❞✐❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✼✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦ Pr♦❞✉t♦ ■♥t❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✽ ❈♦♦r❞❡♥❛❞❛s ❞♦ P♦♥t♦ ❉✐✈✐s♦r ❞❡ ✉♠ ❙❡❣♠❡♥t♦ ❞❡ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✾ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ❚rês P♦♥t♦s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✶✵ ❇❛r✐❝❡♥tr♦ ❞❡ ✉♠ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✶✶ ❊q✉❛çã♦ ❞❛ ❘❡t❛ ♥♦ P❧❛♥♦ ❡ ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✶✶✳✶ ❊q✉❛çã♦ ❈❛rt❡s✐❛♥❛ ❞❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✶✳✶✸ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✶✹ ❊q✉❛çã♦ ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✶✺ ❊q✉❛çã♦ ❞♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✶✻ ❱♦❧✉♠❡ ❞❡ ✉♠ P❛r❛❧❡❧❡♣í♣❡❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✶✼ ❙✐st❡♠❛s ❞❡ ❊q✉❛çõ❡s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✶✽ ❈♦♠♣❛r❛♥❞♦ ▼ét♦❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✶✳✶✾ ❙✐st❡♠❛ ❈❛rt❡s✐❛♥♦ ❖❜❧íq✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✷ ❆♣❧✐❝❛çõ❡s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✹✵ ✷✳✶ ❆♣❧✐❝❛çõ❡s ♥❛ ●❡♦♠❡tr✐❛ P❧❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷ ❆♣❧✐❝❛çõ❡s ♥❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✳✸ ❆♣❧✐❝❛çõ❡s ♥❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✷✳✹ ❆♣❧✐❝❛çõ❡s ♥❛ ❈✐♥❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✷✳✺ ❖✉tr❛s ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✷✳✻ ◗✉❡stõ❡s ❞❛ ❖❇▼❊P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❉✐stâ❝✐❛s ❊♥tr❡ ♣♦♥t♦s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❆ ❡ ❇ ♥♦ ❊s♣❛ç♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❙❡❣♠❡♥t♦s ❊q✉✐♣♦❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ ❱❡t♦r q✉❡ P❛rt❡ ❞❛ ❖r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❱❡t♦r ❉❡✜♥✐❞♦ ♣♦r ❉♦✐s P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✻ ➶♥❣✉❧♦ ❊♥tr❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✼ ▲❡✐ ❞♦s ❈♦ss❡♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✽ P♦♥t♦s ❆❧✐♥❤❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✾ ❙❡❣♠❡♥t♦s P❡r♣❡♥❞✐❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✶✵ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❘❡t❛s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✶✶ ❘❡t❛s P❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✶✷ P❛r❛❧❡❧♦❣r❛♠♦ ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✶✸ ❘❡tâ♥❣✉❧♦ ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✶✹ ❚r✐â♥❣✉❧♦s ❈♦♥❣r✉❡♥t❡s ❆❇❈ ❡ ❉❈❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✶✺ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❈❡♥tr♦ ❆ ❡ ❘❛✐♦ ❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✶✻ P❧❛♥♦ ❉❡t❡r♠✐♥❛❞♦ ♣♦r ✉♠ P♦♥t♦ ❡ ✉♠ ❱❡t♦r P❡r♣❡♥❞✐❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✶✼ P❛r❛❧❡❧❡♣í♣❡❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✶✽ ❘❡t❛s ❝♦♠ ✉♠ P♦♥t♦ ❡♠ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✶✳✷✶ ❘❡t❛ ♥♦ ❙✐st❡♠❛ ❖❜❧íq✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷✳✷✷ ❘❡♣r❡s❡♥t❛çã♦ ●rá✜❝❛ ❞❡ ✉♠❛ ❘♦t❛ ❞❡ ❈♦❧✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✷✳✷✸ ❉✐♠❡♥sõ❡s ❞♦ ❈❛♠♣♦ ❞❡ ❋✉t❡❜♦❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✷✳✷✹ ❉✐❛❣r❛♠❛ ❞♦ ▲✉❝r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✷✳✷✺ ❉✐✈✐sã♦ ❞❡ ✉♠ ❙❡❣♠❡♥t♦ ❡♠ ❚rês P❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✷✳✷✻ ❚r✐â♥❣✉❧♦ ❝♦♠ ✉♠ ❞♦s ❱ért✐❝❡s ♥❛ ❖r✐❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✳✷✼ ❚r✐â♥❣✉❧♦ ❙❡♠❡❧❤❛♥t❡ ❡♠ ❉❡st❛q✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✳✷✽ ❉✐❛❣r❛♠❛ ❞❡ ❊s♣❛ç♦ ❆♠♦str❛❧ ❡ ❊✈❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✷✳✷✾ ▼❛♣❛ ❞♦ ❚❡s♦✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✷✳✸✵ ▲♦❝❛❧✐③❛çã♦ ❞♦ ❚❡s♦✉r♦ ♥♦ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✷✳✸✶ P♦♥t♦s ❙♦❜r❡ ✉♠ ❈✐r❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✷✳✸✷ ❚r❛♣é③✐♦ ❞❛ ◗✉❡stã♦ ✶✼✱ ❖❇▼❊P ✷✵✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✷✳✸✸ ❚r❛♣é③✐♦ ■♥s❝r✐t♦ ♥♦ ❈✐r❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✷✳✸✹ P♦❧í❣♦♥♦ ❞❛ ◗✉❡stã♦ ✶✽ ❞❛ ❖❇▼❊P ✷✵✵✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✷✳✸✺ P❡rí♠❡tr♦ ❞♦ P♦❧í❣♦♥♦ ❆❇❈● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✷✳✸✻ Pr❡ç♦ ♣♦r ❉✐stâ♥❝✐❛ P❡r❝♦rr✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

❆ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❝♦st✉♠❛ ❛♣❛r❡❝❡r ♥❛ sér✐❡ ✜♥❛❧ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❞❡♣♦✐s ❞❡ ♦ ❛❧✉♥♦ t❡r t✐❞♦ ❜❛st❛♥t❡ ❝♦♥t❛t♦ ❝♦♠ ❛ tr❛❞✐❝✐♦♥❛❧ ●❡♦♠❡tr✐❛ P❧❛♥❛✳ ❊ss❛ ♠✉❞❛♥ç❛ ❞❡ ❣❡♦♠❡tr✐❛ q✉❛s❡ s❡♠♣r❡ é ✉♠ ❜❛❧❞❡ ❞❡ á❣✉❛ ❢r✐❛ ♥❛ ✈✐❞❛ ❞❛s ♣❡rs♦♥❛❣❡♥s ❡♥✈♦❧✈✐❞❛s✱ ♦ ❛❧✉♥♦ ❡ ♦ ♣r♦❢❡ss♦r✳

●❡r❛❧♠❡♥t❡✱ ❝♦♠ ♦ ❞❡❝♦rr❡r ❞❛s ❛✉❧❛s✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ ❝♦♠♦ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧í✲ t✐❝❛ é ❧❡❝✐♦♥❛❞❛✱ ♦ ❛❧✉♥♦ ❛❝❛❜❛ ✈❡♥❞♦ ✉♠ ❛♣❛♥❤❛❞♦ ❞❡ ❢ór♠✉❧❛s ❛ s❡r❡♠ ❞❡❝♦r❛❞❛s✱ ❢ór♠✉❧❛ ❞♦ ♣♦♥t♦ ♠é❞✐♦✱ ❜❛r✐❝❡♥tr♦✱ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✱ ❛s ✈ár✐❛s ❡q✉❛çõ❡s ❞❛ r❡t❛ ✭❣❡r❛❧✱ r❡❞✉③✐❞❛✱ s❡❣♠❡♥tár✐❛✱ ❡t❝✮✱ ❛s ❝♦♥❞✐çõ❡s ❞❡ ♣❛r❛❧❡❧✐s♠♦✱ ♣❡r♣❡♥❞✐✲ ❝✉❧❛r✐s♠♦✱ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦ ❡ r❡t❛ ✭❝✉❥❛ ❢ór♠✉❧❛ r❛r❛♠❡♥t❡ ♦ ♣r♦❢❡ss♦r ❞❡❞✉③✮✱ ❛s ❞✉❛s ❡q✉❛çõ❡s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✭❣❡r❛❧ ❡ r❡❞✉③✐❞❛✮ ❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ t❛♥❣❡♥❝✐❛✲ ♠❡♥t♦✳ ❊✱ ✜♥❛❧♠❡♥t❡✱ ❡ ♠✉✐t♦ r❛r❛♠❡♥t❡ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ❡q✉❛çõ❡s ❞❛s ❝ô♥✐❝❛s✱ ❡❧✐♣s❡✱ ❤✐♣ér❜♦❧❡ ❡ ♣❛rá❜♦❧❛✱ q✉❛♥❞♦ ♦ ❛❧✉♥♦ ❥á ♣♦❞❡ ❡stá ❞❡s♠♦t✐✈❛❞♦ ❡ ♦ ♣r♦❢❡ss♦r ❢r✉st❛❞♦ ♣♦r ♥ã♦ t❡r ❝♦♥s❡❣✉✐❞♦ ❝❛t✐✈❛r ♦ ❛❧✉♥♦✳

❖♥❞❡ ❡st❛r ❛ ❢❛❧❤❛ ❡ ❝♦♠♦ ❛❝❤❛♠♦s q✉❡ ❡ss❛ ❝♦♥❞✐çã♦ ♣♦❞❡ s❡r ♠♦❞✐✜❝❛❞❛ ♣❛r❛ ♠❡❧❤♦r❄ ◆❛ ♥♦ss❛ ♦♣✐♥✐ã♦✱ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❝♦♠♦ ❡stá ♥❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❛♣r❡s❡♥t❛ s✉❛s ❢ór♠✉❧❛s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♦s ❡①❡r❝í❝✐♦s ❝♦♠♦ ♠❡r❛ ❛♣❧✐❝❛çõ❡s ❞✐r❡t❛s ♣❛r❛ s✉❛ ♠❡♠♦r✐③❛çã♦✳ P♦r ❡①❡♠♣❧♦✱ ❝❛❧❝✉❧❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s ✭✶✱✷✮ ❡ ✭✼✱✶✵✮❀ ❝❛❧❝✉❧❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ ✭✶✱✸✮ à r❡t❛ 12x+ 7y+ 1 = 0❀ ❡s❝r❡✈❛

❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ✭✷✱✷✮ ❡ ✭✺✱✸✮✱ ❡t❝✳ ❆ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✜❝❛ ♣❛r❡❝❡♥❞♦ ✉♠❛ ♣❛rt❡ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ s❡ ❡♥❝❡rr❛ ❡♠ s✐ ♠❡s♠❛✳ ◆❛ ♥♦ss❛ ♦♣✐♥✐ã♦✱ é ❛í ♦♥❞❡ s❡ ❡♥❝♦♥tr❛ ❛ ♠❛✐♦r ❢❛❧❤❛ ♥♦ ❡♥s✐♥♦ ❞❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❈❡rt♦ q✉❡ ❞❡✈❡♠♦s ✈❛❧♦r✐③❛r ❛s ❢ór♠✉❧❛s✱ ❛✜♥❛❧ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ é ❜❛s✐❝❛♠❡♥t❡ ✐ss♦✱ ✉♠ ❡st✉❞♦ ❞❡ ●❡♦♠❡tr✐❛ P❧❛♥❛ ❡ ❊s♣❛❝✐❛❧ ♣♦r ♠❡✐♦ ❞❡ ❡q✉❛çõ❡s✳ ▼❛s✱ s❡ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ é ✐ss♦✱ é ♣r✐♠♦r❞✐❛❧ q✉❡ ♥ós ♣r♦❢❡ss♦r❡s✱ r❡✈✐s✐t❡♠♦s ❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✱ ❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ❛ ❈✐♥❡♠át✐❝❛ ❊s❝❛❧❛r✱ ❡t❝✱ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ❡♥✉♥❝✐❛❞♦s ❛♣❛r❡♥t❡♠❡♥t❡ ❡s♣❡❝í✜❝♦s ❞❡ss❡s ❝♦♥t❡ú❞♦s✱ ♣♦ré♠ ✉t✐❧✐③❛♥❞♦ r❡❝✉rs♦s ❛♣r❡♥❞✐❞♦s ❡♠ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❖ ✉s♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣❧❛♥♦ ❡ ♥♦ ❡s♣❛ç♦✱ ♦❢❡r❡❝❡ ♥ã♦ ❛♣❡♥❛s ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s ❝♦♠ r❡❝✉rs♦s ❞❡ á❧❣❡❜r❛ ❝♦♠♦✱ r❡❝✐♣r♦❝❛♠❡♥t❡✱ ❢♦r♥❡❝❡ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ✈❛❧✐♦s❛ ♣❛r❛ q✉❡stõ❡s ❞❡ ♥❛t✉r❡③❛ ❛❧❣é❜r✐❝❛✳ ❆ ❛❜♦r❞❛❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ q✉❡ ❢❛r❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦✱ ❧✐♠✐t❛✲s❡ ❛ ✉t✐❧✐③❛✲❧á ♥❛ r❡s♦❧✉çã♦ ❞❡ s✐t✉❛çõ❡s ♣r♦❜❧❡♠❛✱ ♦♥❞❡ ♣♦ss❛ ❣❡r❛r s♦❧✉çõ❡s ♠❛✐s s✐♠♣❧❡s✱ ❞♦ q✉❡ ❛s ❤❛❜✐t✉❛❧♠❡♥t❡ ✉s❛❞❛s ♣❛r❛ r❡s♦❧✈❡r ❡ss❡s ♣r♦❜❧❡♠❛s✳

❆ ♠❡t♦❞♦❧♦❣✐❛ ♣r♦♣♦st❛ ♥❡st❛ ❞✐ss❡rt❛çã♦ é ❛♠♣❛r❛❞❛ ♥♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉✲ ❧❛r❡s ◆❛❝✐♦♥❛✐s✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❡♠ ❝♦♥s♦♥â♥❝✐❛ ❝♦♠ ♦ tr❛♥s❝r✐t♦ ❛❜❛✐①♦✿

✧➱ ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✉♠ t❡♠❛ ♣❡r♠✐t✐r ❝♦♥❡①õ❡s ❡♥tr❡ ❞✐✈❡rs♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❡ ❡♥tr❡ ❞✐✈❡rs❛s ❢♦r♠❛s ❞❡ ♣❡♥s❛♠❡♥t♦ ♠❛t❡♠át✐❝♦✧✭P❈◆✱ ♣❛❣✳ ✹✵✮✳

❈♦♠ ❛ ♠❡t♦❞♦❧♦❣✐❛ ♣r♦♣♦st❛ ❛♣r♦①✐♠❛♠♦s ♠❛✐s ❛✐♥❞❛ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❞❛s ●❡♦♠❡tr✐❛s P❧❛♥❛ ❡ ❊s♣❛❝✐❛❧ ❡ ❞❡ ♦✉tr♦s ❝♦♥t❡ú❞♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ■ss♦ t♦r♥❛ ♣♦ssí✈❡❧ ❢❛③❡r ✉♠❛ ❝♦♥❡①ã♦ ❝♦♠ ♦✉tr♦s ❝♦♠ r❛♠♦s ❞❛ ❝✐ê♥❝✐❛✱ ❢❛❝✐❧✐t❛♥❞♦ ❛ss✐♠✱ às ❝♦♥t❡①t✉❛❧✐③❛çõ❡s ❡ ❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡ ❡ ♠♦str❛♥❞♦ ❛s ✈ár✐❛s ✉t✐❧✐❞❛❞❡s ❞❛

❖ ❈❛♣ít✉❧♦ ✶ ✐♥tr♦❞✉③ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡q✉✐♣♦❧ê♥❝✐❛ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s q✉❡ s❡rá ❛ ❜❛s❡ ♣❛r❛ ❛ ✐♥tr♦❞✉çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r✳ ❆❜♦r❞❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ✈❡t♦r ♥♦ ❡s♣❛ç♦✱ ❢❛③❡♥❞♦ ♦ ❡st✉❞♦ ❞❛s ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ✈❡t♦r❡s✱ ❛ ❛❞✐çã♦✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ❡s❝❛❧❛r✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ❡ ❞❡✜♥✐çã♦ ❞❡ ✈❡t♦r ♥♦ ♣❧❛♥♦✳ ■♥tr♦❞✉③ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❛❜♦r❞❛ ♦s ❝♦♥❝❡✐t♦s ❡ ❢ór♠✉❧❛s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ P❧❛♥❛ ❡ ❊s♣❛❝✐❛❧✳ ➱ ✐♠♣♦rt❛♥t❡ ❞❡✐①❛r ❝❧❛r♦ q✉❡ ❛ ❛❜♦r❞❛❣❡♠ ❢❡✐t❛ ♥❡st❡ ❝❛♣ít✉❧♦ ♥ã♦ s❡rá ❛♣r♦❢✉♥❞❛❞❛✱ ♣♦✐s s❡rá ❛❜♦r❞❛❞♦ ❛♣❡♥❛s ❝♦♥❝❡✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❈❛♣ít✉❧♦ ✷✳

❖ ❈❛♣ít✉❧♦ ✷ tr❛t❛ ❞❛s ❛♣❧✐❝❛çõ❡s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦✲ ❜❧❡♠❛s✱ q✉❡ ♥♦ s❡✉ ❝♦♥t❡①t♦ ❣❡r❛❧ ♥ã♦ ❝✐t❛ ❝♦♦r❞❡♥❛❞❛s✳

❖s r❡❣✐str♦s ❣❡♦♠étr✐❝♦s ♣r❡s❡♥t❡s ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦ ❢♦r❛♠ ♦❜t✐❞♦s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✳ ✸

✸_{●❡♦●❡❜r❛ ✭❛❣❧✉t✐♥❛çã♦ ❞❛s ♣❛❧❛✈r❛s ●❡♦♠❡tr✐❛ ❡ ➪❧❣❡❜r❛✮ é ✉♠ ❛♣❧✐❝❛t✐✈♦ ❞❡ ▼❛t❡♠át✐❝❛}

❉✐♥â♠✐❝❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦s ✈ár✐♦s ♥í✈❡✐s ❞❡ ❡♥s✐♥♦✳ ❖ s♦❢t✇❛r❡ r❡ú♥❡ r❡❝✉rs♦s ❞❡ ●❡♦♠❡tr✐❛✱ ➪❧❣❡❜r❛✱ t❛❜❡❧❛s✱ ❣rá✜❝♦s✱ Pr♦❜❛❜✐❧✐❞❛❞❡✱ ❊st❛tíst✐❝❛ ❡ ❝á❧❝✉❧♦s s✐♠❜ó❧✐❝♦s ❡♠ ✉♠ ú♥✐❝♦ ❛♠❜✐❡♥t❡✳ ❋♦✐ ❝r✐❛❞♦ ♣❡❧♦ ❛✉strí❛❝♦ ▼❛r❦✉s ❍♦❤❡♥✇❛rt❡r ♣❛r❛ s❡r ✉t✐❧✐③❛❞♦ ❡♠ ❛♠❜✐❡♥t❡ ❞❡ s❛❧❛ ❞❡ ❛✉❧❛✳

❈❛♣ít✉❧♦ ✶

❱❡t♦r❡s ❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛

◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❡st❛❜❡❧❡❝❡r ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ✈❡t♦✲ r❡s ❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ❱❛♠♦s ❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s✱ ♠✉❧t✐✲ ♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ✈❡t♦r❡s ❡♠ r❡❧❛çã♦ ❛♦s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ♦rt♦❣♦♥❛✐s ✜①♦s OXY ❡OXY Z✱ ❛❜♦r❞❛r❡♠♦s ❛s ❛♣❧✐✲

❝❛çõ❡s ❞♦s ✈❡t♦r❡s ❡♠ ❛❧❣✉♥s ❝♦♥t❡ú❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ P❧❛♥❛ ❡ ❊s♣❛❝✐❛❧ ❡ ❞❡♠♦str❛r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❡①♣r❡ssõ❡s ❡ t❡♦r❡♠❛s ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ◆♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦ ♠♦str❛r❡♠♦s ❛❧❣✉♠❛s ❞❛s ♣r✐♥❝✐♣❛✐s ✈❛♥t❛❣❡♥s ❞♦ ✉s♦ ❞❡ ✈❡t♦r❡s ♥❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ♦ ♠ét♦❞♦ ❛♥❛❧ít✐❝♦ tr❛❞✐❝✐♦♥❛❧✳

✶✳✶ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ♥♦ P❧❛♥♦

❙❡❥❛♠ P = (x1, y1) ❡Q= (x2, y2) ♣♦♥t♦s ♥♦ ♣❧❛♥♦π ❞❛❞♦s ♣❡❧❛s s✉❛s ❝♦♦r❞❡♥❛✲

❞❛s ❡♠ r❡❧❛çã♦ ❛ ✉♠ s✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s OXY ❞❛❞♦✳ ❆ ❞✐stâ♥❝✐❛ ❞❡ P ❛ Q✱ q✉❡ ❞❡s✐❣♥❛♠♦s ♣♦r d(P,Q)✱ é ❛ ♠❡❞✐❞❛ ❞❛ ❤✐♣♦t❡♥✉s❛ P Q❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦

△P QR✱ ❞❛ ❋✐❣✉r❛ ✶✳✶✱ ❞❡ ❝❛t❡t♦sP R ❡QR✳ ❙❡♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡

❋✐❣✉r❛ ✶✳✶✿ ❉✐stâ❝✐❛s ❊♥tr❡ ♣♦♥t♦s ♥♦ P❧❛♥♦

✉♠ ❡✐①♦ ♠❡❞✐❞❛ ♣❡❧♦ ♠ó❞✉❧♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s✱ ❛s ♠❡❞✐❞❛s ❞❡ss❡s ❝❛t❡t♦s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ |P R_| = _|x2−x1| ❡ |QR| = |y2−y1|✳ ❉♦ t❡♦r❡♠❛ ❞❡

P✐tá❣♦r❛s✱ ♦❜t❡♠♦s

d(P,Q) =|P Q|=

p

|P R_|2_+|QR|2 ₌p_(x

2−x1)2 + (y2−y1)2.

❙❡M é ✉♠ ♣♦♥t♦ ❞♦ s❡❣♠❡♥t♦ P Q t❛❧ q✉❡ d(M,Q) =d(M,P)✱ ❞✐r❡♠♦s q✉❡M é ♦

♣♦♥t♦ ♠é❞✐♦ ❞❡ P Q✱ ♦♥❞❡M = P +Q

2 ✱ q✉❡ ❞❡♠♦str❛r❡♠♦s ♥❛ ❙❡çã♦ ✶✳✽✳

✶✳✷ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ♥♦ ❊s♣❛ç♦

❉❛❞♦s ♦s ♣♦♥t♦s A = (z, y, z) ❡ B = (x′

, y′

, z′

) ♣❡rt❡♥❝❡♥t❡s ❛♦ ❡s♣❛ç♦ R3✳

❈❤❛♠❛✲s❡ ❞✐stâ♥❝✐❛ ❡♥tr❡ A ❡B ❡ ❞❡♥♦t❛♠♦s ♣♦rd(A,B)✱ ♦ ♥ú♠❡r♦ r❡❛❧ ❝♦rr❡s♣♦♥❞❡

❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AB q✉❡ é ❝❛❧❝✉❧❛❞♦ ❞❛ ❢♦r♠❛ d(A,B) =

p

(x₋x′

)2_{+ (y−y}′

)2_{+ (z−z}′

)2_.

✶✳✸✳ ❊◗❯■P❖▲✃◆❈■❆ ❉❊ ❙❊●▼❊◆❚❖❙ ❖❘■❊◆❚❆❉❖❙

❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ P = (x, y, z)à ♦r✐❣❡♠ O = (0,0,0)é ❞❛❞❛ ♣♦r

d(O,P)=

p

x2_+y2_+z2✳ P♦r ♦✉tr♦ ❧❛❞♦ ♣♦❞❡♠♦s ♣r♦✈❛r ❢❛❝✐❧♠❡♥t❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡

A ❡ B✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦ AP B ❞❛ ❋✐❣✉r❛ ✶✳✷✳

❋✐❣✉r❛ ✶✳✷✿ ❉✐stâ♥❝✐❛ ❊♥tr❡ ❆ ❡ ❇ ♥♦ ❊s♣❛ç♦

✶✳✸ ❊q✉✐♣♦❧ê♥❝✐❛ ❞❡ ❙❡❣♠❡♥t♦s ❖r✐❡♥t❛❞♦s

❚❛♥t♦ ♥♦ ♣❧❛♥♦ ❝♦♠♦ ♥♦ ❡s♣❛ç♦✱ ❞✐r❡♠♦s q✉❡ ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦s

AB ❡ CD sã♦ ❡q✉✐♣♦❧❡♥t❡s ❡ ❞❡♥♦t❛♠♦s ♣♦r AB _≡CD✱ q✉❛♥❞♦ ❡❧❡s✿ t❡♠ ♦ ♠❡s♠♦

❝♦♠♣r✐♠❡♥t♦❀ sã♦ ♣❛r❛❧❡❧♦s ♦✉ ❝♦❧✐♥❡❛r❡s ❡ t❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✳ ❙❡ AB ❡ CD sã♦

s❡❣♠❡♥t♦s ♣❛r❛❧❡❧♦s ❡ ❞❡ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ❡♥tã♦AB ❡CDtê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦

❋✐❣✉r❛ ✶✳✸✿ ❙❡❣♠❡♥t♦s ❊q✉✐♣♦❧❡♥t❡s

q✉❛♥❞♦ABCD é ✉♠❛ ♣❛r❛❧❡❧♦❣r❛♠♦✳ P❛r❛ q✉❡ ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦sAB

❡CDs❡❥❛♠ ❡q✉✐♣♦❧❡♥t❡s é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ ♦s s❡❣♠❡♥t♦sAD❡BC t❡♥❤❛♠

♦ ♠❡s♠♦ ♣♦♥t♦ ♠é❞✐♦✳

❱❛♠♦s ❝❛r❛❝t❡r✐③❛r ❛ ❡q✉✐♣♦❧ê♥❝✐❛ ❡♠ t❡r♠♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ P❛r❛ ✐ss♦✱ ❝♦♥✲ s✐❞❡r❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s ❖❳❨ ♥♦ ♣❧❛♥♦✱ ❡ s❡❥❛♠ A = (x1, y1)✱

B = (x2, y2)✱ C = (x3, y3)❡D= (x4, y4) ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ❡①♣r❡ss♦s ❡♠ ❝♦♦r❞❡♥❛❞❛s

❝♦♠ r❡❧❛çã♦ ❛♦ s✐st❡♠❛ ❞❛❞♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳ AB _≡CD _⇐⇒ x2−x1 =x4−x3 ❡ y2−y1 =y4−y3✳

P❛r❛ ✈❡r✐✜❝❛r ❡ss❛ ♣r♦♣♦s✐çã♦✱ ❞❡✈❡♠♦s ❧❡♠❜r❛r q✉❡ AD ❡ BC t❡♠ ♦ ♠❡s♠♦

♣♦♥t♦ ♠é❞✐♦✱ ❡♥tã♦

AB _≡CD _⇐⇒

x1+x4

2 ,

y1+y4

2

=

x2 +x3

2 ,

y2+y3

2

⇐⇒ (x1+x4, y1+y4) = (x2+x3, y2+y3)

⇐⇒ x1+x4 =x2+x3 −→y1+y4 =y2+y3

⇐⇒ x2−x1 =x4−x3 −→y2−y1 =y4−y3.

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦str❛r✳

❊①❡♠♣❧♦ ✶✳✶✳ ❉❛❞♦s ♦s ♣♦♥t♦s A= (1,2)✱ B = (3,₋2) ❡ C = (₋2,0), ❞❡t❡r♠✐♥❡

❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ D= (x, y) ❞❡ ♠♦❞♦ q✉❡ AB _≡CD✳

P❡❧❛ ♣r♦♣♦s✐çã♦ ✶✳✶✱ s❡AB _≡CD ❡♥tã♦ 3₋2 =x₋(₋2) ❡₋2₋2 =y₋0q✉❡

♥♦s ❞❛r x=₋1 ❡ y=₋4✱ ♦✉ s❡❥❛ D= (₋1,₋4)✳

✶✳✹✳ ❱❊❚❖❘❊❙ ◆❖ ❊❙P❆➬❖

✶✳✹ ❱❡t♦r❡s ♥♦ ❊s♣❛ç♦

◗✉❛♥❞♦ ♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦sAA′ _{❡P P}′ _{♥♦ ❡s♣❛ç♦E sã♦ ❡q✉✐♣♦❧❡♥t❡s✱ ❡s❝r❡✲} ✈❡♠♦s −−→AA′ ₌ −−→_{P P}′ _{❡ ❞✐③❡♠♦s q✉❡ ❡❧❡s r❡♣r❡s❡♥t❛♠ ♦ ♠❡s♠♦ ✈❡t♦r~v =}−−→_AA′ ₌ −−→_{P P}′_✳ ❉❛❞♦ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s OXY Z✱ ❝♦♠ A = (a, b, c)✱ A′ _{= (a}′_{, b}′_{, c}′_{)✱ P =}

(m, n, p)❡P′ _{= (m}′_{, n}′_{, p}′_{)✱ t❡♠✲s❡}−−→_AA′ ₌−−→_{P P}′ _{=~v s❡✱ ❡ s♦♠❡♥t❡ s❡✱a}′_−a=m′_−m✱

b′

−b =n′

−n ❡c′

−c=p′

−p✳ P♦♥❞♦α =a′

−a✱β =b′

−b ❡γ =c′

−c✱ ❡s❝r❡✈❡✲s❡ ~v = (α, β, γ) ❡ ❞✐③✲s❡ q✉❡ ❡st❛s sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r ~v = −−→AA′ _{♥♦ s✐st❡♠❛}

OXY Z✳ ❙❡ ~v = −−→AA′ _{é ✉♠ ✈❡t♦r ❡ P é ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ♥♦ ❡s♣❛ç♦✱ ❡①✐st❡ ✉♠} ú♥✐❝♦ ♣♦♥t♦ P′ _{t❛❧ q✉❡} −−→_{P P}′ _{= ~v✳ ◗✉❛♥❞♦ P = (x, y, z) ❡ ~v = (α, β, γ)✱ t❡♠✲s❡}

P′ _{= (x+α, y+β, z+γ)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡♥❞♦O = (0,0,0)❡ P = (x, y, z)♣♦♥t♦s} ❞♦ ❡s♣❛ç♦✱ ♦ ✈❡t♦r~v =−→OP = (x, y, z)✳

❋✐❣✉r❛ ✶✳✹✿ ❱❡t♦r q✉❡ P❛rt❡ ❞❛ ❖r✐❣❡♠

❆ ✐❣✉❛❧❞❛❞❡ ❞❡ ✈❡t♦r❡s é ❡st❛❜❡❧❡❝✐❞❛ ❝♦♠ ❜❛s❡ ♥❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡ ~u = (x1, y1, z1) ❡ ~v = (x2, y2, z2)✱ ❡♥tã♦ ~u =~v ⇐⇒ x1 = x2✱

y1 = y2 ❡ z1 = z2✳ ❯♠❛ ❝♦✐s❛ ❛❣r❛❞á✈❡❧ ❝♦♠ r❡s♣❡✐t♦ ❛ ✈❡t♦r❡s é q✉❡ s❡ ♣♦❞❡♠

❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❡♥tr❡ ❡❧❡s✳ ❆s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❛s ♦♣❡r❛çõ❡s t♦r♥❛♠✲s❡ ♣❛rt✐❝✉✲ ❧❛r♠❡♥t❡ s✐♠♣❧❡s s❡ ❝♦♥✈❡♥❝✐♦♥❛r♠♦s ❡♠ ❛❞♠✐t✐r ♦ ✈❡t♦r ♥✉❧♦ −→AA✱ ❞❡t❡r♠✐♥❛❞♦ ♣♦r

✉♠ s❡❣♠❡♥t♦ ❞❡❣❡♥❡r❛❞♦✱ ♥♦ q✉❛❧ ♦ ✐♥í❝✐♦ ❡ ❛ ❡①tr❡♠✐❞❛❞❡ ✜♥❛❧ s❡ r❡❞✉③❡♠ ❛ ✉♠ ♠❡s♠♦ ♣♦♥t♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛❞♠✐t✐r❡♠♦s q✉❡ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r ❞♦ ❡s✲ ♣❛ç♦ sã♦ ❡q✉✐♣♦❧❡♥t❡s❀ ❛ss✐♠ ♦ ✈❡t♦r ♥✉❧♦ −→AA ♣♦❞❡ t❡r✱ ❝♦♠♦ ♦s ❞❡♠❛✐s ✈❡t♦r❡s✱

✉♠❛ ♦r✐❣❡♠ ❧♦❝❛❧✐③❛❞❛ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞♦ ❡s♣❛ç♦✳ ❯s❛r❡♠♦s ♦ ♠❡s♠♦ s✐♠❜♦❧♦0

♣❛r❛ r❡♣r❡s❡♥t❛r t❛♥t♦ ♦ ✈❡t♦r ♥✉❧♦ ❝♦♠♦ ♦ ♥ú♠❡r♦ ③❡r♦✳ ❊♠ q✉❛❧q✉❡r s✐st❡♠❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r ♥✉❧♦ sã♦ (0,0,0)✳

❙❡❥❛♠A= (x1, y1, z1)✱B = (x2, y2, z2)♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❡O ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛

❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ✜❝❛ ❡st❛❜❡❧❡❝✐❞❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r −→AB ❞❛

s❡❣✉✐♥t❡ ❢♦r♠❛✿ −→OA+−→AB =−−→OB✱ ♦✉ s❡❥❛A+−→AB =B =_⇒−→AB =B₋A✱ ✈❡r ❋✐❣✉r❛

✶✳✺✳ ❉❛í t❡♠♦s q✉❡ ♦ ✈❡t♦r −→AB = (x2−x1, y2−y1, z2−z1)✳

❋✐❣✉r❛ ✶✳✺✿ ❱❡t♦r ❉❡✜♥✐❞♦ ♣♦r ❉♦✐s P♦♥t♦s

✶✳✺✳ ❖P❊❘❆➬Õ❊❙ ❈❖▼ ❱❊❚❖❘❊❙ ◆❖ ❊❙P❆➬❖

✶✳✺ ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ♥♦ ❊s♣❛ç♦

❙❡❥❛♠ ~u = (x1, y1, z1) ❡~v = (x2, y2, z2) ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❡①♣r❡ss♦s ❡♠ ❝♦♦r❞❡✲

♥❛❞❛s✳ ❱❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ s♦♠❛ ❞♦s ✈❡t♦r❡s ~u ❡ ~v ❡♠ ❢✉♥çã♦

❞❡ s✉❛s ❝♦♦r❞❡♥❛❞❛s✳ ❙❡❥❛♠ A = (x1, y1, z1) ❡ B = (x2, y2, z2) t❛✐s q✉❡ ~u = −→OA ❡

~v=−−→OB✳ ❙❡❥❛C= (x3, y3, z3)♦ ♣♦♥t♦ t❛❧ q✉❡~u =−−→BC✳ ❖s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦sOA

❡ BC sã♦ ❡q✉✐♣♦❧❡♥t❡s✱ ❧♦❣♦ ♦s ✈❡t♦r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s −→OA ❡−−→BC sã♦ ✐❣✉❛✐s✱ ✐st♦ é

(x1, y1, z1) = (x3−x2, y3−y2, z3−z2)✳ ▲♦❣♦✱C = (x3, y3, z3) = (x1+x2, y1+y2, z1+z2)✳

❆ss✐♠✱ ~u+~v =−→OA+OB−−→=−−→OB +−−→BC =−→OC =C = (x1+x2, y1+y2, z1+z2)✳

P❛r❛ ❛s ♦♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ♥♦ ♣❧❛♥♦✱ ♦ ♣r♦❝❡ss♦ é ❛♥á❧♦❣♦ ❛♦ ❡❢❡t✉❛❞♦ ♣❛r❛ ❞❡✜♥✐r ❡st❛s ♦♣❡r❛çõ❡s ♣❛r❛ ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❜❛s✐❝❛♠❡♥t❡ ❛s ♠❡s♠❛s✱ ♣♦r ✐ss♦ ♠✉✐t♦s ❞❡t❛❧❤❡s s❡rã♦ ♦♠✐t✐❞♦s✳ ❱❛♠♦s ❞❡✜♥✐r ❞✉❛s ♦♣❡r❛çõ❡s ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s ❞♦ ❡s♣❛ç♦✱ ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❡ ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r❡s ♣♦r ♥ú♠❡r♦s r❡❛✐s✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❈❤❛♠❛✲s❡ s♦♠❛ ❞❡ ❞♦✐s ✈❡t♦r❡s~u= (x1, y1, z1) ❡~v = (x2, y2, z2) q✉❡

s❡ ✐♥❞✐❝❛ ♣♦r~u+~v✱ ♦ ✈❡t♦r✿ ~u+~v = (x1, y1, z1)+(x2, y2, z2) = (x1+x2, y1+y2, z1+z2)✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❉❛❞♦ ✉♠ ✈❡t♦r ~v = (x, y, z) ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ µ✱ ❝❤❛♠❛✲s❡ ♣r♦❞✉t♦

❞♦ ✈❡t♦r~v ♣❡❧♦ ❡s❝❛❧❛r µ✱ ♦ ✈❡t♦r ~u=µ~v=µ(x, y, z) = (µx, µy, µz)✳

❖ ✈❡t♦r~u =µ~v t❡rá ♦ ♠❡s♠♦ s❡♥t✐❞♦ ❞❡~v s❡ µ >0 ❡ s❡♥t✐❞♦ ❝♦♥trár✐♦ ❛♦~v✱ s❡ µ <0✳

❊①❡♠♣❧♦ ✶✳✷✳ ❙❡❥❛♠ A = (3,2,0)✱ B = (0,3,₋2) ❡ C = (4,3,2) ♣♦♥t♦s ❞♦ ❡s♣❛ç♦✳

❖❜t❡♥❤❛ ♦ ♣♦♥t♦ D t❛❧ q✉❡ −−→AD=−→AB+−→AC✳

❙♦❧✉çã♦✿ ❚❡♠♦s✱ −→AB = (0₋3,3₋2,₋2₋0) = (₋3,1,₋2) ❡ −→AC = (4₋3,3₋ 2,2₋0) = (1,1,2)✳ ▲♦❣♦✱−→AB+−→AC = (₋3,1,₋2)+(1,1,2) = (₋2,2,0)✳ ❆❧é♠ ❞✐ss♦✱

s❡ D = (x4, y4, z4) é ❛ ❡①tr❡♠✐❞❛❞❡ ❞♦ r❡♣r❡s❡♥t❛♥t❡ AD ❞♦ ✈❡t♦r s♦♠❛ −→AB +−→AC

❝♦♠ ♦r✐❣❡♠ ♥♦ ♣♦♥t♦ A✱ ❡♥tã♦ x4 −3 = −2✱ y4 −2 = 2 ❡ z4 −0 = 0✳ P♦rt❛♥t♦✱

D= (1,4,0)✳

✶✳✻ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s

❆ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r❡s ♣♦r ❡s❝❛❧❛r❡s s❛t✐s❢❛③❡♠ ♣r♦✲ ♣r✐❡❞❛❞❡s s❡♠❡❧❤❛♥t❡s às ♣r♦♣r✐❡❞❛❞❡s ❛r✐t♠ét✐❝❛s ❞❛s ♦♣❡r❛çõ❡s ♥✉♠ér✐❝❛s✱ ✐st♦ ♣❡r♠✐t❡ ❝♦♥✈❡rt❡r ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s ❡♠ ♣r♦❜❧❡♠❛s ❛❧❣é❜r✐❝♦s ❡ ✈✐❝❡ ✲ ✈❡rs❛✳

✶✳✻✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ❆❞✐çã♦ ❞❡ ❱❡t♦r❡s

❙❡~u, ~v ❡ w~ sã♦ ✈❡t♦r❡s ❞♦ ❡s♣❛ç♦✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

❛✮ ~u+~v=~v+~u ✭❈♦♠✉t❛t✐✈❛✮❀

❜✮ ~u+ (~v+w) = (~u+~v) +w~ ✭❆ss♦❝✐❛t✐✈❛✮❀

❝✮ ~u+~0 = ~u ✭❡❧❡♠❡♥t♦ ♥❡✉tr♦✮❀

❞✮ ~u+ (₋~u) =~0 ✭❊①✐stê♥❝✐❛ ❞❡ ✐♥✈❡rs♦ ❛❞✐t✐✈♦✮✳

❆ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s é ✉♠❛ ♦♣❡r❛çã♦ ❜❡♠ ❞❡✜♥✐❞❛✱ ✐st♦ é✱ ❛ ❞❡✜♥✐çã♦ ❞❛ s♦♠❛ ❞♦ ✈❡t♦r ~u=−→AB ❝♦♠~v =−−→BC ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣♦♥t♦ ❆✳

P❛r❛ ❝❛❞❛ ✈❡t♦r~u✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✈❡t♦r✱ q✉❡ ❞❡s✐❣♥❛♠♦s ♣♦r ₋~u✱ ❞❡♥♦♠✐♥❛❞♦

s✐♠étr✐❝♦ ❛❞✐t✐✈♦ ❞❡ ~u✱ t❛❧ q✉❡ ❛ s♦♠❛ ❞❡ ❛♠❜♦s é ♦ ✈❡t♦r ♥✉❧♦✳ ❖ ✐♥✈❡rs♦ ❛❞✐t✐✈♦

❞❡ ~v = (x, y, z) é ♦ ✈❡t♦r ₋~v = (₋x,₋y,₋z)✳ P❛r❛ ♣r♦✈❛r ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❛ ♣❛rt❡

✶✳✻✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆❙ ❖P❊❘❆➬Õ❊❙ ❈❖▼ ❱❊❚❖❘❊❙

b)✱ ❝♦♥s✐❞❡r❛♠♦s ~u= (x1, y1, z1)✱~v = (x2, y2, z2) ❡w~ = (x3, y3, z3)✳ ❊♥tã♦✱

~u+ (~v+w~) = (x1, y1, z1) + [(x2, y2, z2) + (x3, y3, z3)]

= (x1, y1, z1) + (x2+y2+z2, x3+y3+z3)

= [x1+ (x2+x3), y1+ (y2+y3), z1+ (Z2+z3)]

= [(x1+x2) +x3,(y1+y2) +y3,(z1+z2) +z3]

= (x1+x2, y1+y2, z1+z2) + (x3, y3, z3)

= [(x1, y1, z1) + (x2, y2, z2)] + (x3, y3, z3)

= (~v+~u) +w.~

❆s ♣r♦✈❛s ❛❧❣é❜r✐❝❛s ❞❛s ♣❛rt❡s ❛✮ ❡ ❝✮ sã♦ ❛♥á❧♦❣❛s✳

✶✳✻✳✷ Pr♦♣r✐❡❞❛❞❡s ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❱❡t♦r❡s ♣♦r ❊s❝❛❧❛r❡s

❙❡~u ❡~v sã♦ ✈❡t♦r❡s ❡λ ❡ µ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s

❛✮ λ(µ_·~v) = (λµ)~v ✭❆ss♦❝✐❛t✐✈✐❞❛❞❡✮❀

❜✮ (λ+µ)~v =λ~v+µ~v ✭❉✐str✐❜✉t✐✈❛ ❡♠ r❡❧❛çã♦ ❛ ❛❞✐çã♦ ❞❡ ✈❡t♦r❡s✮❀

❝✮ λ(~u+~v) =λ~u+λ~v ✭❉✐str✐❜✉t✐✈❛ ❝♦♠ r❡❧❛çã♦ ❛ ❛❞✐çã♦ ❞❡ ❡s❝❛❧❛r❡s✮❀

❞✮ 1~v =~v ✭❊❧❡♠❡♥t♦ ♥❡✉tr♦✮✳

❆ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐str✐❜✉t✐✈❛s sã♦ ✈❡r✐✜❝❛❞❛s ✉s❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s q✉❡ ❥á ❝♦♥❤❡❝❡♠♦s ♥♦s ♥ú♠❡r♦s r❡❛✐s✳ ❉❡♠♦str❛♥❞♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ♦ ✐t❡♠ c)✳ ❙❡❥❛♠ ~u= (x1, y1, z1)❡~v = (x2, y2, z2) ✈❡t♦r❡s ❞♦ ❡s♣❛ç♦

❡ λ_∈R✳ t❡♠♦s q✉❡

λ(~u+~v) = λ[(x1, y1, z1) + (x2, y2, z2)]

= λ(x1 +x2, y1+y2, z1+z2)

= (λx1 +λx2, λy1+λy2, λz1+λz2)

= (λx1, λy1, λz1) + (λx2, λy2, λz2)

= λ(x1, y1, z1) +λ(x2, y2, z2)

= λ~u+λ~v,

♦ q✉❡ ♠♦str❛ ❛ ♣r♦♣r✐❡❞❛❞❡✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ sã♦ ❢❡✐t❛s ❛s ♣r♦✈❛s ❛❧❣é❜r✐❝❛s ❞❛s ♣❛rt❡s ❛✮✱ ❜✮ ❡ ❞✮✳

✶✳✼ Pr♦❞✉t♦ ■♥t❡r♥♦ ❡♠ ❚❡r♠♦s ❞❡ ❈♦♦r❞❡♥❛❞❛s

❈❤❛♠❛✲s❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ ♦s ✈❡t♦r❡s ~u = (x1, y1, z1) ❡ ~v = (x2, y2, z2) ♦

♥ú♠❡r♦ r❡❛❧ h~u, ~v_i = x1x2 +y1y2 +z1z2✳ ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ❞♦✐s ✈❡t♦r❡s é ✉♠

♥ú♠❡r♦ r❡❛❧ ❡ ♥ã♦ ✉♠ ✈❡t♦r✳ ❆❧❣✉♠❛s ✈❡③❡s é ❝❤❛♠❛❞♦ ❞❡ ♣r♦❞✉t♦ ❡s❝❛❧❛r✳ ❊st❛ ❢ór♠✉❧❛ ✈❛❧❡ ♦❜✈✐❛♠❡♥t❡ q✉❛♥❞♦ ✉♠ ❞♦s ✈❡t♦r❡s ~u ♦✉ ~v é ✐❣✉❛❧ ❛ ③❡r♦✳ ❆ss✐♠✱

❡♠ q✉❛❧q✉❡r ❝❛s♦✱ ♦❜t❡♠♦s ✉♠❛ ❡①♣r❡ssã♦ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h~u, ~v_i ❡♠ ❢✉♥çã♦

❞❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈❡t♦r❡s ~u ❡ ~v✳ P❛r❛ ❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ♣r❡❝✐s❛♠♦s ❞❡

❞♦✐s ❝♦♥❝❡✐t♦s ♣r❡❧✐♠✐♥❛r❡s✱ ❛ ♥♦çã♦ ❞❡ ♥♦r♠❛ ❞❡ ✉♠ ✈❡t♦r ❡ ❛ ♥♦çã♦ ❞❡ â♥❣✉❧♦ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s✳ ❉❡✜♥✐♠♦s ♦ â♥❣✉❧♦ ❡♥tr❡ ~u ❡ ~v ❝♦♠♦ s❡♥❞♦ ♦ ♠❡♥♦r â♥❣✉❧♦

❡♥tr❡ ♦s s❡❣♠❡♥t♦s AB ❡ AC r❡♣r❡s❡♥t❛♥t❡s ❞❡ ~u ❡~v✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❋✐❣✉r❛ ✶✳✻✳

❉❡♥♦t❛♠♦s ♣♦r θ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❡♥tr❡~u ❡~v✳

❊①❡♠♣❧♦ ✶✳✸✳ ❉❡t❡r♠✐♥❡ ♦ ♥ú♠❡r♦ r❡❛❧ x ♣❛r❛ q✉❡ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞♦s ✈❡t♦r❡s

✶✳✼✳ P❘❖❉❯❚❖ ■◆❚❊❘◆❖ ❊▼ ❚❊❘▼❖❙ ❉❊ ❈❖❖❘❉❊◆❆❉❆❙

❋✐❣✉r❛ ✶✳✻✿ ➶♥❣✉❧♦ ❊♥tr❡ ❱❡t♦r❡s

~u= (4,₋3,0) ❡~v = (x,1,1) s❡❥❛ ✐❣✉❛❧ ❛ ✺✳

❚❡♠♦s✿ 5 =_h~u, ~v_i= 4x₋3_·1 + 0_·1_⇐⇒8 = 4x_⇐⇒x= 2✳

✶✳✼✳✶ ◆♦r♠❛ ❞❡ ✉♠ ❱❡t♦r

❆ ♥♦r♠❛ ♦✉ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r~v é ♦ ♥ú♠❡r♦_k~v_k❞❛❞♦ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦

❞❡ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ r❡♣r❡s❡♥t❛♥t❡ ❞❡~v✳ ❙❡ A = (x1, y1, z1) ❡ B = (x2, y2, z2)

sã♦ ❞♦✐s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦✱ ❡♥tã♦ ❛ ♥♦r♠❛ ❞♦ ✈❡t♦r~v =−→AB é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡A❡B✳

❈♦♠♦ −→AB = (x2−x1, y2−y1, z2−z1)✱ s❡❣✉❡✱ ❞❡♥♦t❛♥❞♦ ♣♦rd ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ A❡

B✱ q✉❡_k~v_k=d=p

(x2−x1)2+ (y2−y1)2+ (z2−z1)✳ ❙❡P = (x, y, z)é ✉♠ ♣♦♥t♦

t❛❧ q✉❡ ~v = −→OP✱ ❡♥tã♦ d(O, P) = _k~v_k = px2_+y2_+z2✳ ❙❡ _{~u = (x, y, z)}✱ ❡♥tã♦ ♦

❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r λ~u é _|λ_| ✈❡③❡s ❛ ♥♦r♠❛ ❞♦ ✈❡t♦r ~u✱ ♦✉ s❡❥❛✱ _kλ~u_k =_|λ_|._k~uk✳

❉❡ ❢❛t♦✱ s❡❥❛ ~u = (x, y, z) ❡λ_∈R✳ ❆ss✐♠✱ λ~u= (λx, λy, λz)❡✱ ♣♦rt❛♥t♦

kλ~u_k = p(λx)2_{+ (λy)}2_{+ (λz}2₎

= pλ2_x2_+λ2_y2 _+λ2_z2

= pλ2_(x2_+y2_+z2₎

= √λ2_.p_x2_+y2_+z2

= _|λ_|._k~u_k.

✶✳✼✳✷ ❱❡t♦r❡s P❡r♣❡♥❞✐❝✉❧❛r❡s

Pr♦♣♦s✐çã♦ ✶✳✷✳ ❉♦✐s ✈❡t♦r❡s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ✭❞❡t❡r♠✐♥❛♠ ✉♠ â♥❣✉❧♦ r❡t♦✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ s❡✉ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ③❡r♦✳

❈♦♥s✐❞❡r❡♠♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ♦rt♦❣♦♥❛❧ ♦s ♣♦♥t♦sO = (0,0,0)✱A= (x1, y1, z1)✱

B = (x2, y2, z2)✱ ♦ ✈❡t♦r−→OA= (x1, y1, z1)❡ ♦ ✈❡t♦r−−→OB = (x2, y2, z2)✱ ❝♦♠ −→OA⊥−−→OB✱

❛♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❞❡✈❡♠♦s t❡r(OA)2_{+ (OB)}2 _{= (AB)}2 _{⇒ k}−→_OAk2₊

k−−→OB_k2 _=k−→_ABk2_{✱ ♦✉ s❡❥❛x}2

1+y12+z12+x22+y22+z22 = (x2−x1)2+(y2−y1)2+(z2−z1)2

s✐♠♣❧✐✜❝❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s 0 = ₋2x1x2−2y1y2−2z1z2✱ q✉❡ ♥♦s ❞❛r

x1x2+y1y2+z1z2 = 0✱ q✉❡ ♣r♦✈❛ ❛ ♣r♦♣♦s✐çã♦✳

➱ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r ❛ r❡❧❛çã♦ ❞❡ ♦rt♦❣♦♥❛❧✐❞❛❞❡ ❡♥tr❡ ♦s ✈❡t♦r❡s ~u = (a, b) ❡

~v = (₋b, a)✳ ❈♦♠♦ _h~u, ~v_i = 0 ❡♥tã♦ ❡❧❡s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳ ❆❧é♠ ❞✐ss♦✱ _k~u_k =

k~v_k=√a2_+b2 _{❡ ♣♦rt❛♥t♦✱ ❡❧❡s ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱~v é ♦ r❡s✉❧t❛❞♦ ❞❛}

r♦t❛çã♦ ❞❡ ~u✱ ❞❡ ✉♠ â♥❣✉❧♦ r❡t♦ ✭♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦✱ ✐st♦ é✱ ♠❡s♠♦ s❡♥t✐❞♦ ❞❡OX

♣❛r❛ OY✮✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦ ✈❡t♦r ~v′ _{= (a,}

−b)✱ é ♦ r❡s✉❧t❛❞♦ ❞❛ r♦t❛çã♦ ❞❡ ~u✱ ❞❡

✉♠ â♥❣✉❧♦ r❡t♦✱ ♥♦ s❡♥t✐❞♦ ❛♥t❡ ❤♦rár✐♦✳

✶✳✼✳ P❘❖❉❯❚❖ ■◆❚❊❘◆❖ ❊▼ ❚❊❘▼❖❙ ❉❊ ❈❖❖❘❉❊◆❆❉❆❙

✶✳✼✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦ Pr♦❞✉t♦ ■♥t❡r♥♦

❙❡ ~u✱ ~v ❡ w~ sã♦ ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❡ k é ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❡♥tã♦ ❛s s❡❣✉✐♥t❡s

♣r♦♣r✐❡❞❛❞❡s ✈❛❧❡♠✿

❛✮ h~u, ~v_i=_h~v, ~ui❀

❜✮ h~u,(~v+w~)_i=_h~u, ~v_i+_h~u, ~w_i❀

❝✮ k(_h~u, ~v_i) = _h(k~u), ~v_i=_h~u,(k~v)_i❀

❞✮ h~u, ~u_i=_k~u_k2 _≥0✳

P❛r❛ ❞❡♠♦str❛r ♦ ✐t❡♠ a)✱ s❡❥❛♠ ~u = (x1, y1, z1) ❡ ~v = (x2, y2, z2)✳ ❊♥tã♦ h~u, ~vi =

x1x2+y1y2+z1z2 =x2x1+y2y1+z2z1 =h~v, ~ui✳ P❛r❛ ♦ ✐t❡♠b)✱ s❡❥❛♠~u= (x1, y1, z1)✱

~v= (x2, y2, z2)❡ w~ = (x3, y3, z3)✳ ❆ss✐♠ t❡♠♦s

h~u, ~v+w~_i = _h(x1, y1, z1),[(x2, y2, z2) + (x3, y3, z3)]i

= _h(x1, y1, z1),(x2+x3, y2+y3, z2+z3)i

= x1(x2+x3) +y1(y2+y3) +z1(z2+z3)

= x1x2+x1x3+y1y2+y1y3+z1z2+z1z3

= (x1x2+y1y2+z1z2) + (x1x3 +y1y3+z1z3)

= _h~u, ~v_i+_h~u, ~w_i.

P❛r❛ s❛❜❡r ♠❛✐s✱ ♠❡❞✐r ♦ â♥❣✉❧♦ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s ❞♦ ❡s♣❛ç♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞❡t❡r✲ ♠✐♥❛r ♦ s❡✉ ❝♦ss❡♥♦✱ ♣♦✐s ♦ â♥❣✉❧♦✱ q✉❛♥❞♦ ♠❡❞✐❞♦ ❡♠ r❛❞✐❛♥♦s✱ é ✉♠ ♥ú♠❡r♦ ❞♦ ✐♥t❡r✈❛❧♦ [0, π]❡ ♦ ❝♦ss❡♥♦ r❡str✐t♦ ❛ ❡ss❡ ✐♥t❡r✈❛❧♦ é ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛✳

P❛r❛ ♣r♦✈❛r ❛ ♣❛rt❡c)❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ❝♦♥s✐❞❡r❛♥❞♦k ✉♠❛

❝♦♥st❛♥t❡ r❡❛❧✱ t❡♠♦s

k_h~u, ~v_i = k[_h(x1, y1, z1),(x2, y2, z2)i]

= k(x1x2+y1y2+z1z2)

= k(x1x2) +k(y1y2) +k(z1z2)

= (kx1)x2+ (ky1)y2+ (kz1)z2

= _h(kx1, ky1, kz1),(x2, y2, z2)i

= _h(k~u), ~v_i.

❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣♦ ♠♦str❛✲s❡ q✉❡ k_h~u, ~v_i = _h~u,(k~v)_i✳ P❛r❛ ❞❡♠♦str❛r ❛ ♣❛rt❡ d)✱

❝♦♥s✐❞❡r❛♥❞♦~u = (x, y, z)✱ t❡♠♦s_h~u, ~u_i=x2_+y2_+z2 _=k~uk2 _≥0✳

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ♦✉ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ♦s ✈❡t♦r❡s ~u ❡ ~v✱ é ♦

♥ú♠❡r♦ r❡❛❧ h~u, ~v_i=_k~u_kk~v_kcosθ✱ ♦♥❞❡ θ=∠(~u, ~v) ❡ s❡✱~u= 0 ♦✉ ~v = 0✱ _h~u, ~v_i= 0✳

❋✐❣✉r❛ ✶✳✼✿ ▲❡✐ ❞♦s ❈♦ss❡♥♦s

P❛r❛ ✈❡r✐✜❝❛r ❡st❛ ♣r♦♣♦s✐çã♦✱ t♦♠❡♠♦s −→OP = ~u = (x1, y1, z1) ❡ −→OQ = ~v =

(x2, y2, z2) ♥♦ tr✐â♥❣✉❧♦ OP Q ❞❛ ❋✐❣✉r❛ ✶✳✼ ❡ ❛♣❧✐❝❛♠♦s ❛ ❧❡✐ ❞♦s ❝♦ss❡♥♦s✱ ♦ q✉❡

✶✳✼✳ P❘❖❉❯❚❖ ■◆❚❊❘◆❖ ❊▼ ❚❊❘▼❖❙ ❉❊ ❈❖❖❘❉❊◆❆❉❆❙

r❡s✉❧t❛ ♥❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿

k−→QP_k2 _=k~uk2_+k~vk2_{−2k~ukk~vkcosθ (1)}

P♦r ♦✉tr♦ ❧❛❞♦−→OQ+−→QP =−→OP =_⇒−→QP =−→OP ₋−→OQ✱ ❡♥tã♦

k−→QP_k2 = _k−→OP ₋−→OQ_k2 =_k~u₋~v_k2

= _k(x1−x2, y1−y2, z1−z2)k2

= (x1−x2)2+ (y1 −y2)2+ (z1−z2)2

= x2₁+y2₁ +z₁2+x2₂+y₂2+z2_{2 −}2(x1x2+y1y2+z1z2). ✭✶✳✶✮

❙✉❜st✐t✉✐♥❞♦ (1.1) ❡♠ (1)✱ r❡s✉❧t❛

k~u_kk~v_kcosθ=x1x2+y1y2+z1z2 =h~u, ~vi

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦str❛r✳

Pr♦♣♦s✐çã♦ ✶✳✹✳ P❛r❛ t♦❞♦s ♦s ✈❡t♦r❡s ~u ❡ ~v✱ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲

❙❝❤❛✇❛r③✱

|h~u, ~v_{i| ≤ k}~u_kk~v_k.

P❛r❛ ♣r♦✈❛r ❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ t♦♠❛♥❞♦ ♦ ♠ó❞✉❧♦ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ✐❞❡♥t✐❞❛❞❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✱ t❡♠♦s |h~u, ~v_i| =_|k~u_k._k~v_k|cosθ_| = _k~u_k._k~v_k|cosθ|✳ ❙❛✲

❜❡♥❞♦ q✉❡ |cosθ_{| ≤}1✱ ✈❡♠ _|h~u, ~v_{i| ≤ k}~u_k._k~vk✳

Pr♦♣♦s✐çã♦ ✶✳✺✳ P❛r❛ t♦❞♦s ♦s ✈❡t♦r❡s ~u ❡ ~v✱ ❞♦ ❡s♣❛ç♦ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥✲

❣✉❧❛r✱ k~u+~v_{k ≤ k}~u_k+_k~v_k✱ ✈❛❧❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉♠ ❞♦s ✈❡t♦r❡s ~u

♦✉ ~v é ③❡r♦ ♦✉ sã♦ ♠ú❧t✐♣❧♦s ♣♦s✐t✐✈♦s ✉♠ ❞♦ ♦✉tr♦✳

❉❡♠♦str❛çã♦✳ ❈♦♠♦ ❛s q✉❛♥t✐❞❛❞❡s ❞❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ sã♦ t♦❞❛s ♥ú♠❡r♦s r❡❛✐s

♥ã♦ ♥❡❣❛t✐✈♦s✱ ❡❧❛ ❡q✉✐✈❛❧❡ à ❞❡s✐❣✉❛❧❞❛❞❡

k~u+~v_k2 _≤(_k~u_k+_k~v_k)2.

❉❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ t❡♠♦s

k~u+~v_k2 = _h~u+~v, ~u+~v_i

= _h~u, ~u_i+_h~u, ~v_i+_h~v, ~u_i+_h~v, ~v_i

= _ku˘_k2+ 2_h~u, ~v_i+_k~v_k2

≤ k~u_k2+ 2_k~u_kk~v_k+_k~v_k2

= (_k~u_k+_k~v_k)2.

✶✳✽ ❈♦♦r❞❡♥❛❞❛s ❞♦ P♦♥t♦ ❉✐✈✐s♦r ❞❡ ✉♠ ❙❡❣♠❡♥t♦

❞❡ ❘❡t❛

❆ ❢ór♠✉❧❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ♣❡r♠✐t❡ ♦❜t❡r ✐♠❡❞✐❛t❛♠❡♥t❡ ❛s ❝♦♦r❞❡✲ ♥❛❞❛s ❞♦ ♣♦♥t♦ q✉❡ ❞✐✈✐❞❡ ✉♠ s❡❣♠❡♥t♦AA′ _{♥✉♠❛ r❛③ã♦ ❞❛❞❛✳ ❙❡♥❞♦A= (a, b, c)❡}

A′ _{= (a}′_{, b}′_{, c}′_{)✳ ❱❡r❡♠♦s q✉❡ ♦s ♣♦♥t♦s ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛AA}′ _sã♦X

t= (xt, yt, zt)✱

♦♥❞❡ 0_≤t_≤1 ❡

xt=a+t(a′−a)

yt =b+t(b′−b)

zt=c+t(c′−c).

✶✳✾✳ ❆▲■◆❍❆▼❊◆❚❖ ❉❊ ❚❘✃❙ P❖◆❚❖❙ ◆❖ P▲❆◆❖

❉❡st❛s ✐❣✉❛❧❞❛❞❡s✱ r❡s✉❧t❛ ♣♦r ✉♠ ❝á❧❝✉❧♦ s✐♠♣❧❡s q✉❡

d(A, Xt)

d(A, A′₎ =

p

(xt−a)2+ (yt−b)2+ (zt−c)2 p

(a′_−a)2_{+ (b}′_−b)2_{+ (c}′_−c)2 =t.

P♦rt❛♥t♦✱ Xt é✱ ♣❛r❛ t♦❞♦ t ∈ [0,1]✱ ♦ ♣♦♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AA′ t❛❧ q✉❡

d(A, Xt)/d(A, A′) = t✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t♦♠❛♥❞♦ t =

1

2 ♦❜t❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦

♣♦♥t♦ ♠é❞✐♦ ❞❡ AA′

M =X1/2 =

a+a′

2 ,

b+b′

2 ,

c+c′

2

.

✶✳✾ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ❚rês P♦♥t♦s ♥♦ P❧❛♥♦

❉✐③❡♠♦s q✉❡ três ♣♦♥t♦s ❞✐st✐♥t♦s ❡stã♦ ❛❧✐♥❤❛❞♦s ♦✉ sã♦ ❝♦❧✐♥❡❛r❡s✱ q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s três✳ P❛r❛ q✉❡ A = (x1, y1)✱ B = (x2, y2) ❡ C =

❋✐❣✉r❛ ✶✳✽✿ P♦♥t♦s ❆❧✐♥❤❛❞♦s

(x3, y3)❡st❡❥❛♠ ❛❧✐♥❤❛❞♦s é ♣r❡❝✐s♦ q✉❡ ♦s ✈❡t♦r❡s−→AB ❡−→AC s❡❥❛♠ ❝♦❧✐♥❡❛r❡s✱ ✐st♦ é✿

−→

AB =α.−→AC✱ ♣❛r❛ ❛❧❣✉♠α_∈R✳ ❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s✱

t❡♠✲s❡ (x2−x1, y2−y1) =α(x3−x1, y3 −y1)⇒

x2−x1

x3−x1

= y2−y1

y3−y1

=α✱ ✉s❛♥❞♦ ❛

♥♦t❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

x2−x1 y2−y1

x3−x1 y3−y1

= 0.

❖❜s❡r✈❡ q✉❡ ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❝♦rr❡s♣♦♥❞❡ às ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r −→

AB ❡ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ às ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r−→AC✳

✶✳✶✵ ❇❛r✐❝❡♥tr♦ ❞❡ ✉♠ ❚r✐â♥❣✉❧♦

❇❛r✐❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ♦ ♣♦♥t♦ ♦♥❞❡ ❛s r❡t❛s s✉♣♦rt❡s ❞❛s três ♠❡❞✐❛♥❛s s❡ ✐♥t❡rs❡❝t❛♠✳ ▲❡♠❜r❛♥❞♦ q✉❡ ✉♠❛ ♠❡❞✐❛♥❛ é ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❧✐❣❛ ✉♠ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦✱ ❛♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ ♦♣♦st♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ♥♦ ♣❧❛♥♦ ♦s ♣♦♥t♦s

A= (x1, y1)✱B = (x2, y2)❡C = (x3, y3)❞✐st✐♥t♦s ❡ ♥ã♦ ❝♦❧✐♥❡❛r❡s✳ ❙❡❥❛G= (xG, yG)

♦ ❜❛r✐❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳ ❖❜s❡r✈❛♠♦s q✉❡✱ s❡♥❞♦ M = (xM, yM) ♣♦♥t♦

♠é❞✐♦ ❞❡ AC✱ t❡♠✲s❡✿ xM =

x1+x3

2 ❡ yM =

y1+y3

2 ✳ ❖ ♣♦♥t♦ G é t❛❧ q✉❡

−−→

BG = 2−−→GM✱ ❞❛✐✿ G₋B = 2(M ₋G) _⇒ 3G = 2M +B _⇒ G = 2M+B

3 ⇒

(xG, yG) =

2(xM, yM) + (x2, y2)

3 ✳ ❉❛✐✱ t❡r❡♠♦s

xG =

2xM +x2

3 =

2

x1+x3

2

+x2

3 =

x1+x2+x3

3 ,

yG =

2yM +y2

3 =

2

y1+y3

2

+y2

3 =

y1+y2 +y3

3

✶✳✶✶✳ ❊◗❯❆➬➹❖ ❉❆ ❘❊❚❆ ◆❖ P▲❆◆❖ ❊ ◆❖ ❊❙P❆➬❖

❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❜❛r✐❝❡♥tr♦ sã♦

G=

x1+x2+x3

3 ,

y1+y2+y3

3

.

✶✳✶✶ ❊q✉❛çã♦ ❞❛ ❘❡t❛ ♥♦ P❧❛♥♦ ❡ ♥♦ ❊s♣❛ç♦

❙❡ r é ✉♠❛ r❡t❛ ♥♦ ♣❧❛♥♦✱ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ A = (a, b) ❡ t❡♠ ❞✐r❡çã♦ ❞❡

~v = (α, β)₆=~0✱ s❡♥❞♦ P = (x, y)_∈ r t❡♠♦s P _∈r _⇐⇒−→AP é ♠ú❧t✐♣❧♦ ❞❡ ~v✳ ❊♥tã♦✱ P _∈ r _⇐⇒ AP = t.~v ♣❛r❛ ❛❧❣✉♠ t _∈ R✱ ❞❛✐ P = A+t.~v✱ ❝♦♠ t _∈ R✳ ▲♦❣♦

(x, y) = (a, b) +t(α, β)✳ P♦rt❛♥t♦

r:     

x=a+αt

y =b+βt, t _∈R.

❊st❛s sã♦ ♣♦rt❛♥t♦✱ ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦A= (a, b)

❡ t❡♠ ❛ ❞✐r❡çã♦ ❞♦ ❞♦ ✈❡t♦r~v = (α, β)✳

❊①❡♠♣❧♦ ✶✳✹✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s A = (1,2) ❡

B = (4,4)✳

❙❡♥❞♦ P = (x, y) ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦✱ s♦❜r❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ❡♠ A ❡ B✳ ❚❡♠♦s

q✉❡✿ −→AB = ~v = (4₋1,4₋2) = (3,2) ❡ −→AP = t.~v✱ ❝♦♠ t _∈ R✱ ♣♦✐s ~v ❡ −→AP sã♦

❝♦❧✐♥❡❛r❡s✳ ❡♥tã♦ P ₋A=t.~v=_⇒P =A+t.~v=_⇒(x, y) = (1,2) +t(3,2)✱ ♦✉ s❡❥❛

r:     

x= 1 + 3t

y= 2 + 2t, t_∈R.

❊st❛s sã♦ ♣♦rt❛♥t♦✱ ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛ r❡t❛ q✉❡ ❝♦♥té♠ ♦s ♣♦♥t♦sA= (1,2)

❡ B = (4,4)✳ ❈♦❧♦❝❛♥❞♦ t ❡♠ ❢✉♥çã♦ ❞❡ x ❡ y✱ t❡♠♦s✿ x−1

3 =

y₋2

2 =⇒2x−2 = 3y₋6 =_⇒2x₋3y+ 4 = 0✳

◆♦ ❡s♣❛ç♦✱ ✉♠❛ r❡t❛ r ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ A = (a, b, c) ❡ t❡♠ ❞✐r❡çã♦ ❞❡ ~v = (α, β, γ)₆=~0✱ s❡♥❞♦ P = (x, y, z)_∈r t❡♠♦s P _∈r_⇐⇒−→AP é ♠ú❧t✐♣❧♦ ❞❡ ~v✳ ❊♥tã♦✱ P _∈ r _⇐⇒ −→AP = t.~v ♣❛r❛ ❛❧❣✉♠ t _∈ R✱ ❞❛✐ P = A+t.~v✱ ❝♦♠ t _∈ R✳ ▲♦❣♦

(x, y, z) = (a, b, c) +t(α, β, γ)✳ P♦rt❛♥t♦

r :



    

    

x=a+αt y=b+βt

z =c+γt, t_∈R.

❊st❛s sã♦ ♣♦rt❛♥t♦✱ ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ A = (a, b, c) ❡ t❡♠ ❛ ❞✐r❡çã♦ ❞♦ ❞♦ ✈❡t♦r~v= (α, β, γ)✳

✶✳✶✶✳✶ ❊q✉❛çã♦ ❈❛rt❡s✐❛♥❛ ❞❛ ❘❡t❛

P❛r❛ ❝❛r❛❝t❡r✐③❛r ❛ ❡q✉❛çã♦ ❝❛rt❡s✐❛♥❛ ❞❡ ✉♠❛ r❡t❛✱ ✈❛♠♦s ✉s❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ♣❡r♣❡♥❞✐❝✉❧❛r✐❞❛❞❡ ❞❡ ❞♦✐s ✈❡t♦r❡s✱ ✐st♦ é✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ ❡❧❡s é ✐❣✉❛❧ ❛ ③❡r♦✳ ❯♠ ✈❡t♦r ~u ₆= 0 é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ✉♠❛ r❡t❛ r s❡ ♦s ✈❡t♦r❡s ~u ❡ −→AB sã♦

♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♣♦♥t♦s A✱ B _∈ r✳ ❙❡❥❛ r ❛ r❡t❛ q✉❡ ♣❛ss❛

♣❡❧♦ ♣♦♥t♦ A = (x0, y0) ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ✈❡t♦r ~v = (a, b) 6= 0✳ P❛r❛ q✉❛❧q✉❡r

♣♦♥t♦P = (x, y)❞❛ r❡t❛r✱ t❡r❡♠♦s q✉❡_h−→AP , ~v_i= 0✳ ❊♥tã♦✱a(x₋x0)+b(y−y0) = 0

❞❛í✱ ax+by =ax0+by0✳ ❋❛③❡♥❞♦ c=ax0+by0 t❡♠♦s

ax+by =c.