VETORES: UMA ABORDAGEM PARA O ENSINO MÉDIO.

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ▼❆❘❆◆❍➹❖

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ✲ P❘❖❋▼❆❚

P❆❚❘❮❈■❖ ❚❖❘❘❊❙ ❈❖❙❚❆

❱❊❚❖❘❊❙✿ ❯▼❆ ❆❇❖❘❉❆●❊▼ P❆❘❆ ❖ ❊◆❙■◆❖ ▼➱❉■❖

P❆❚❘❮❈■❖ ❚❖❘❘❊❙ ❈❖❙❚❆

❱❡t♦r❡s✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛✲ ♥❤ã♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❆♥tô♥✐♦ P✐r❡s ❋❡rr❡✐r❛ ▼❛rã♦

❊❧❛❜♦r❛❞❛ ♣❡❧❛ ❇✐❜❧✐♦t❡❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦

❈♦st❛✱ P❛trí❝✐♦ ❚♦rr❡s

❱❡t♦r❡s✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ✴ P❛trí❝✐♦ ❚♦rr❡s ❈♦st❛✳ ❙ã♦ ▲✉✐s✱ ✷✵✶✺✳

✺✾❢✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❆♥tô♥✐♦ P✐r❡s ❋❡rr❡✐r❛ ▼❛rã♦

❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦✱ Pr♦✲ ❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✺

✶✳ ❱❡t♦r❡s ✷✳ ❖♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s✳ ✸✳ ❆♣❧✐❝❛çõ❡s ✈❡t♦r✐❛s ♥❛ ❣❡♦♠❡tr✐❛✳ ✹✳❆♣❧✐❝❛çõ❡s ✈❡t♦r✐❛✐s ♥❛ ❋ís✐❝❛✳ ■✳ ❚ít✉❧♦✳

❆ ♣r❡s❡♥t❡ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋▼❆✴P❘❖❋▼❆❚✱ ❡ ❡❧❛❜♦r❛❞❛ ♣♦r P❛trí❝✐♦ ❚♦rr❡s ❈♦st❛✱ s♦❜ ♦ tít✉❧♦✿ ✏❱❡t♦r❡s✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦

♠é❞✐♦✑✱ ❢♦✐ ❛♣r♦✈❛❞❛ ❡♠ ❞❡ ❞❡

✷✵✶✺ ❡ r❡❝❡❜❡✉ ❝♦♥❝❡✐t♦✿

❈❖▼■❙❙➹❖ ❊❳❆▼■◆❆❉❖❘❆

Pr♦❢✳ ❉r✳ ❏♦sé ❆♥tô♥✐♦ P✐r❡s ❋❡rr❡✐r❛ ▼❛rã♦ ❖r✐❡♥t❛❞♦r

Pr♦❢♦ _{❉r✳ ❘♦❜❡rt♦ ❇❛t✐st❛ ❞♦s ❙❛♥t♦s}

Pr♦❢♦ _{❉r✳ ❏♦ã♦ ❈♦❡❧❤♦ ❙✐❧✈❛ ❋✐❧❤♦}

❉❊❉■❈❆❚Ó❘■❆

❆●❘❆❉❊❈■▼❊◆❚❖❙

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛♦ s❡♥❤♦r ♥♦ss♦ ❉❊❯❙ ♣❡❧❛ s❛❜❡❞♦r✐❛ ♠❡ ❝♦♥❝❡❞✐❞❛ ❞✉r❛♥t❡ t♦❞❛ ❡ss❛ ❝❛♠✐♥❤❛❞❛✳

❆ ♠✐♥❤❛ ❡s♣♦s❛ ❏♦r❝✐❧❛♥❡ ❞❛ ❙✐❧✈❛ ❙♦❛r❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦

❆ ♠❡✉s ♣❛✐s ❘❛✐♠✉♥❞❛ ❚♦rr❡s ❈♦st❛ ❡ ❊❞✉❛r❞♦ r♦❞r✐❣✉❡s ❞❛ ❈♦st❛ ♣♦r s❡♠♣r❡ ❛♣♦✐✲ ❛r❡♠ ❡ ❛❝r❡❞✐t❛r❡♠ q✉❡ ❛tr❛✈és ❞❛ ❡❞✉❝❛çã♦ ♠❡✉ ❢✉t✉r♦ ❡st❛r✐❛ ❣❛r❛♥t✐❞♦✳

❆♦s ❝♦❧❡❣❛s ❞❛ t✉r♠❛ ❞♦ Pr♦❢♠❛t✴❯❢♠❛ ✷✵✶✸✱ ♣❡❧♦ r❡s♣❡✐t♦✱ ❝♦♥s✐❞❡r❛çã♦ ❡ ❛❝♦❧❤✐✲ ♠❡♥t♦ ♥♦s ♠♦♠❡♥t♦s ♥❡❝❡ssár✐♦s ❞♦ ❝✉rs♦✳

❆♦s ❛♠✐❣♦s ❘❛✐♠✉♥❞♦ ❏♦sé P✐♥t♦ ❈✉tr✐♠✱ ●✐❧❢r❛✐♥❡ ❞❛ ❙✐❧✈❛ ❆♠♦r✐♠✱ ❊✈❛❧❞♦ ❞❡ ❆❧✲ ♠❡✐❞❛ ❆❧❜✉q✉❡rq✉❡✱ ▼ár❝✐♦ ❆♥tô♥✐♦ ▼♦t❛ P✐♥t♦✱ ❊❞✐♥❛❧❞♦ ❋♦♥s❡❝❛ ❈♦rr❡❛ ❡ ▼❛r❝♦s ❆♥t♦♥✐♦ ❞❡ ❙♦✉s❛ ❙✐❧✈❛ ♣❡❧❛ ❛♠✐③❛❞❡✱ r❡s♣❡✐t♦✱ ❝♦♥s✐❞❡r❛çã♦ ❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠♣❛rt✐✲ ❧❤❛❞♦ ❞✉r❛♥t❡ ♦ ❝✉rs♦✳

❆♦ ❝♦♦r❞❡♥❛❞♦r ❞♦ ❝✉rs♦ Pr♦❢✳ ❉r✳ ❏♦ã♦ ❞❡ ❉❡✉s ▼❡♥❞❡s✱ q✉❡ ♥ã♦ ♠❡❞✐✉ ❡s❢♦rç♦s ♣❛r❛ ❛❥✉❞❛r ❛ t✉r♠❛✳

❊P❮●❘❆❋❊

❍♦♥r❛ ❛ t❡✉ ♣❛✐ ❡ ❛ t✉❛ ♠ã❡✱ ♣❛r❛ q✉❡ s❡ ♣r♦❧♦♥❣✉❡♠ ♦s t❡✉s ❞✐❛s ♥❛ t❡rr❛ q✉❡ ♦ ❙❡♥❤♦r t❡✉ ❉❊❯❙ t❡ ❞á✳

❘❊❙❯▼❖

❖ ❡st✉❞♦ ❞♦s ✈❡t♦r❡s é ❛♣r❡s❡♥t❛❞♦ ❜✉s❝❛♥❞♦ ♠♦str❛r ♦ t❡♠❛ ♥❛s ❛✉❧❛s ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❋ís✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❯♠ ♠♦❞❡❧♦ ❝♦♥❝❡✐t✉❛❧ q✉❡ é ♣♦st✉❧❛❞♦ ❡ ❢✉♥❞❛♠❡♥t❛❞♦ ♥❛s ♣✉❜❧✐❝❛çõ❡s s♦❜r❡ ✈❡t♦r❡s❀ ♦✉ s❡❥❛✱ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ❝á❧❝✉❧♦ ✈❡t♦r✐❛❧ ❡ á❧❣❡❜r❛ ❧✐✲ ♥❡❛r✳ ■♥tr♦❞✉③✐r❡♠♦s ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❞❡ ♠♦❞♦ ❝♦♥str✉t✐✈♦ ❡ ♣❡❧❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦ ♠❡s♠♦✱ ❞❡♥tr♦ ❞♦ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❊♠ s❡❣✉✐❞❛✱ ✉t✐❧✐③❛♠♦s ♦ ♠ét♦❞♦ ❛♥❛❧ít✐❝♦ ❡ ❣❡♦♠étr✐❝♦ ♣❛r❛ ✐♥tr♦❞✉③✐r ♦✉tr♦s ❝♦♥❝❡✐t♦s ❡ ♦♣❡r❛çõ❡s✳ ◆♦s ♣r❡♦❝✉♣❛♠♦s ❡♠ ❡①♣ôr ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ♥❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❡s♣❛❝✐❛❧ ❝♦♠♦ ❞❡✲ ♠♦str❛çõ❡s ❞❡ ❛❧❣✉♥s t❡♦r❡♠❛s ❡ ♣r♦❜❧♠❡❛s ❝❧áss✐❝♦s ❝♦♠♦ ❝á❝❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❡ ár❡❛s ❛tr❛✈és ❞❡ ✈❡t♦r❡s ❡ t❛♠❜é♠ ♥❛ ár❡❛ ❞❡ ❢ís✐❝❛ ❝♦♠♦ ♦ ❈á❝❧✉❧♦ ❞♦ tr❛❜❛❧❤♦ ❡ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ q✉❡ sã♦ ♦❜t✐❞♦s ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ✈❡t♦r❡s✳

❆❇❙❚❘❆❈❚

❚❤❡ st✉❞② ♦❢ ✈❡❝t♦rs ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ❤❡r❡ ✐♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ s✉❜❥❡❝t ✐♥ ❝❧❛ss❡s ♦❢ ♠❛t❤❡♠❛t✐❝s ❛♥❞ ♣❤②s✐❝s ✐♥ ❤✐❣❤ s❝❤♦♦❧ ✳ ❆ ❝♦♥❝❡♣t✉❛❧ ♠♦❞❡❧ t♦ ❜❡ ♣♦st✉❧❛t❡❞ ❛♥❞ ❜❛s❡❞ ♦♥ ♣✉❜❧✐❝❛t✐♦♥s ♦♥ ✈❡❝t♦rs ❀ ✐✳❡ t❡①t❜♦♦❦s ♦❢ ✈❡❝t♦r ❝❛❧❝✉❧✉s ❛♥❞ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ■♥tr♦❞✉❝❡ t❤❡ st✉❞② ♦❢ ✈❡❝t♦rs ❝♦♥str✉❝t✐✈❡❧② ❣❡♦♠❡tr✐❝❛❧❧② ❛♥❞ t❤❡ ✈✐❡✇ ♦❢ ✐t✱ ✇✐t❤✐♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✳ ❚❤❡♥ ✇❡ ✉s❡ t❤❡ ❛♥❛❧②t✐❝❛❧ ❛♥❞ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞ t♦ ✐♥tr♦❞✉❝❡ ♦t❤❡r ❝♦♥❝❡♣ts ❛♥❞ ♦♣❡r❛t✐♦♥s✳ ❈❛r❡ t♦ ❡①♣♦s❡ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✢❛t ❛♥❞ s♣❛t✐❛❧ ❣❡♦♠❡tr② ❛s ❞❡♠♦♥str❛t✐♦♥s ♦❢ s♦♠❡ ❝❧❛ss✐❝❛❧ t❤❡♦r❡♠s ❛♥❞ ♣r♦❜❧♠❡❛s ❛♥❞ ✈♦❧✉♠❡ ❝á❝❧❝✉❧♦ ❛♥❞ ❛r❡❛s t❤r♦✉❣❤ ✈❡❝t♦rs ❛♥❞ ❛❧s♦ ✐♥ ♣❤②s✐❝s ❛s ❈á❝❧✉❧♦ ❧❛❜♦r ❛♥❞ ♠❛❣♥❡t✐❝ ✜❡❧❞ t❤❛t ❛r❡ ♦❜t❛✐♥❡❞ t❤r♦✉❣❤ t❤❡ ✉s❡ ♦❢ ✈❡❝t♦rs✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❙❡❣✉✐♠❡♥t♦AB ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ P❛r❛❧❡❧❡♣í♣❡❞♦ ABCDEF GH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ ❙❡❣✉✐♠❡♥t♦ ❡q✉✐♣♦❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ABCDEF GH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✻ ❱❡t♦r~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✼ ❱❡t♦r❡s ♦rt♦❣♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✶ ❙♦♠❛ ❞♦s ✈❡t♦r❡s ~u ❡~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ❙♦♠❛ ❞♦s ✈❡t♦r❡s ~u✱~v ❡w~ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❞♦ ✈❡t♦r~v ♣♦r ✭❛✮ α >0✱ ❡ ✭❜✮ ♣♦r α <0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸✳✹ ❱❡t♦r❡s~v✱ 2~v ❡ ₋4~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✺ ❱❡t♦r w~ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡~v ❡~u ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✻ P❛r❛❧❡❧❡♣í♣❡❞♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s~u✱~v ❡ w~ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✼ ❇❛s❡ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✽ ➶♥❣✉❧♦ ❡♥tr❡ ♦s ✈❡t♦r❡s~u ❡~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✾ ❚r✐â♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶✵ Pr♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛E2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✶✶ ➶♥❣✉❧♦ ✐♥s❝r✐t♦ ♥✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✷ Pr♦❞✉t♦ ✈❡t♦r✐❛❧ ❞❡ ~u_×~v ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✸ ❘❡❣r❛ ❞❛ ♠ã♦ ❞✐r❡✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✹ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ♠ó❞✉❧♦ ❞♦ Pr♦❞✉t♦ ❱❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s~u✱~v ❡ w~ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✻ P❛r❛❧❡❧❡♣í♣❡❞♦ ❢♦r♠❛❞♦ ♣❡❧♦s ABCDEF GH ❞❡ ♠❡❞✐❞❛s5x3x2 ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸✳✶✼ ❚❡tr❛❡❞r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✶ ❈♦♠♣♦s✐çã♦ ❞❡ ♠♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✷ ❋♦rç❛ ❆♣❧✐❝❛❞❛ ❛ ✉♠ ❝♦r♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✸ ❋♦rç❛ ❛♣❧✐❝❛❞❛ ❛ ✉♠ ❝♦r♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✹ ❱❡❧♦❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✺ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❋♦rç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✻ ❋♦rç❛ ❊❧étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✶ ❚r✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ABC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✷ ▲❡✐ ❞♦s ❈♦ss❡♥♦sa2

=b2

+c2

−2b_·c_·cosAb ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸ ▲❡✐ ❞♦s ❈♦ss❡♥♦skw~_k2

=_k~u_{k · k}~v_{k ·}cos(θ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✺✳✹ ▲❡✐ ❞♦ ❙❡♥♦ a sin_Ab =

b sin_Bb =

c

sin_Cb ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✺✳✺ ▲❡✐ ❞♦s ❙❡♥♦s kw~_k2

=_k~u_{k · k}~v_{k ·}cos(θ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✺✳✻ ❇❛s❡ ▼é❞✐❛ ❞❡ ✉♠ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✼ ❇❛s❡ ▼é❞✐❛ ❞❡ ✉♠ ❚r✐â♥❣✉❧♦ ❝♦♠ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✻✳✶ ❇❛s❡ ❝❛♥ô♥✐❝❛ ♦rt♦♥♦r♠❛❧ ♣♦s✐t✐✈❛{i, j, k_} ❞♦R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✻✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ♣♦♥t♦s ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

❙❯▼➪❘■❖

✶ ■♥tr♦❞✉çã♦ ✶✵

✷ ■♥tr♦❞✉çã♦ ❛ ➪❧❣❡❜r❛ ❱❡t♦r✐❛❧ ✶✷

✷✳✶ ●r❛♥❞❡③❛s ❊s❝❛❧❛r❡s ❡ ❱❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❙❡❣♠❡♥t♦s ❞❡ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸ ◆♦r♠❛✱ ❉✐r❡çã♦ ❡ ❙❡♥t✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ ❙❡❣✉✐♠❡♥t♦s ❊q✉✐♣♦❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✺ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸ ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ✶✾

✸✳✶ ❙♦♠❛ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❱❡t♦r❡s ♣♦r ✉♠ ❊s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸ ❈♦♠❜✐♥❛çã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✹ ❉❡♣❡♥❞ê♥❝✐❛ ❡ ■♥❞❡♣❡♥❞ê♥❝✐❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✺ ❇❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✻ ❇❛s❡ ❖rt♦❣♦♥❛❧ ❡ ❖rt♦♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✼ ➶♥❣✉❧♦ ❡♥tr❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✽ Pr♦❞✉t♦ ❡♥tr❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✾ Pr♦❞✉t♦ ■♥t❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✶✵ Pr♦❞✉t♦ ❱❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✶ Pr♦❞✉t♦ ▼✐st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✷ ❱♦❧✉♠❡ ❞♦ ❚❡tr❛❡❞r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹ ❆♣❧✐❝❛çõ❡s à ❋ís✐❝❛ ✸✹

✹✳✶ ❈♦♠♣♦s✐çã♦ ❞❡ ▼♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷ ❚r❛❜❛❧❤♦ ❘❡❛❧✐③❛❞♦ ♣♦r ✉♠❛ ❋♦rç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✸ ▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✹ ❈❛♠♣♦ ▼❛❣♥ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✺ ❆♣❧✐❝❛çõ❡s à ●❡♦♠❡tr✐❛ ❡♠ ●❡r❛❧ ✹✵

✺✳✶ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✷ ▲❡✐ ❞♦s ❈♦ss❡♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸ ▲❡✐ ❞♦s ❙❡♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✹ ❇❛s❡ ▼é❞✐❛ ❞❡ ✉♠ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✻ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ♥♦ ❊s♣❛ç♦ ✹✼

✻✳✶ ❱❡t♦r❡s ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✻✳✷ ■❣✉❛❧❞❛❞❡ ❞❡ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✸ ❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✹ ❱❡t♦r ❉❡✜♥✐❞♦ ♣♦r ❉♦✐s P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✻✳✺ ▼ó❞✉❧♦ ❞❡ ✉♠ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻✳✻ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✼ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✸

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✸

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ♠❛t❡r✐❛❧ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ✈❡t♦r❡s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♦♥❞❡ ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s é ❛❜♦r❞❛❞♦ ❛♣❡♥❛s ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❋ís✐❝❛✱ ♠❛s ♣♦ss✉✐ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛✳ ■♠♣❧❡♠❡♥t❛♠♦s ❡ss❡ ❡st✉❞♦ ❢❛③❡♥❞♦ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛tr❛✈és ❞❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡✳ ❆ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡ ♦❢❡r❡❝❡ ✉♠❛ ♥♦✈❛ ❛t✐t✉❞❡ ❞✐❛♥t❡ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ♣♦st✉r❛ ❡♠ ❜✉s❝❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡❧❛ ✈✐s❛ ❣❛r❛♥t✐r ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❧♦❜❛❧✐③❛♥t❡✱ r♦♠♣❡♥❞♦ ❝♦♠ ♦s ❧✐♠✐t❡s ❞❛s ❞✐s❝✐♣❧✐♥❛s✳ ❆ ✉t✐❧✐③❛çã♦ ❞❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❞❡s❡♥✈♦❧✈❡r ✉♠ tr❛❜❛❧❤♦ ❞❡ ✐♥t❡❣r❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s ❞❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❋ís✐❝❛ ❝♦♠ ♦✉tr❛s ár❡❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ é ✉♠❛ ❞❛s ♣r♦♣♦st❛s ❛♣r❡s❡♥t❛❞❛s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❞❛s ❉✐r❡tr✐③❡s ❈✉rr✐❝✉❧❛r❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭❉❈◆❊▼✮ q✉❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞♦ ❛❧✉♥♦✳

❆s ❉❈◆❊▼ ❛rt✐❝✉❧❛♠✲s❡ ❡♠ três ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✿ ▲✐♥❣✉❛❣❡♥s✱ ❈ó❞✐❣♦s ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s❀ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛✱ ▼❛t❡♠át✐❝❛ ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s❀ ❡ ❈✐ê♥❝✐❛s ❍✉♠❛♥❛s ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s✳ ❊❧❛s ❡st❛❜❡❧❡❝❡♠✱ t❛♠❜é♠✱ ❛s ❝♦♠♣❡tê♥❝✐❛s ❡ ❤❛❜✐❧✐❞❛❞❡s q✉❡ ❞❡✈❡rã♦ s❡r✈✐r ❝♦♠♦ r❡❢❡r❡♥❝✐❛✐s ♣❛r❛ ❛s ♣r♦♣♦st❛s ♣❡❞❛❣ó❣✐❝❛s✱ ❛❧é♠ ❞❡ r❡❝♦♠❡♥❞❛r ❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡ ❡ ❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦✱ ♣r✐♥❝í♣✐♦s ❝♦♥❞✉t♦r❡s ❞❛ ♦r❣❛♥✐③❛çã♦ ❝✉rr✐❝✉❧❛r✳ ❖ ❞♦❝✉♠❡♥t♦ ❛♣r❡s❡♥t❛ ❛s ❝♦♥s✐❞❡r❛çõ❡s t❡ór✐❝❛s s♦❜r❡ ❡ss❡s ❞♦✐s ♣r✐♥❝í♣✐♦s ❝♦♠♦ r❡❝✉rs♦s ♣❡❞❛❣ó❣✐❝♦s ♣❛r❛ ✉♠ ❡♥s✐♥♦ q✉❡ ❝♦❧♦q✉❡ ♦ ❛❧✉♥♦ ❝♦♠♦ ❝❡♥tr♦ ❞❡ s✉❛ ❛♣r❡♥❞✐③❛❣❡♠✳

❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡st❡ t❡♠❛ ❢♦✐ ❛ ♦❜s❡r✈❛çã♦ q✉❡ ♦ ❡st✉❞♦ ❞♦s ❱❡t♦r❡s ❞❡s❞❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❋ís✐❝❛✱ é ❢❡✐t♦ ❞❡ ♠♦❞♦ ❞❡s❝♦♥❡①♦ ❡ s❡♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ▼❛t❡♠át✐❝❛ ❝♦♥s✐st❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❜❡ ❞❡st❛❝❛r q✉❡ ♦s ❧✐✈r♦s ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ✉s❛❞♦s ♥♦ ❇r❛s✐❧ ♥ã♦ tr❛❜❛❧❤❛♠ ❝♦♠ ✈❡t♦r❡s✱ ♦ ❡st✉❞♦ é ❢❡✐t♦ ❡ ❛♣r❡s❡♥t❛❞♦ ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❞❛ ❞✐s❝✐♣❧✐♥❛ ❢ís✐❝❛✱ ❡ ♦s ❛❧✉♥♦s t❡♠ ❡①tr❡♠❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❝♦♠♣r❡❡♥❞ê✲❧♦ ♣♦✐s ♦s ✈❡t♦r❡s sã♦ ❛♣r❡s❡♥t❛❞♦s ❞❡ ❢♦r♠❛ ❞✐r❡t❛ s❡♠ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞❡ s✉❛s ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s ❡ ❣❡♦♠étr✐❝❛s✱ ❞❡✐①❛♥❞♦ ❞❡ ❛t❡♥✉❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❛s ❡str✉t✉r❛s ♠❛t❡♠át✐❝❛s ❡ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r é ♣✉r❛♠❡♥t❡ ♠❛t❡♠át✐❝♦✳ ❛ss✐♠✱ ❡ss❛ ♣r♦♣♦st❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❝♦♠ ❡ss❛ ♣❛rt✐❝✉❧❛r✐❞❛❞❡ ❞❡ ❞❡s❡♥✈♦❧✈❡r ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s ❞❡ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ♠❛t❡♠át✐❝❛ s❡♠♣r❡ ❞❛♥❞♦ ❡①❡♠♣❧♦s ❞❛ ✉t✐❧✐③❛çã♦ ♥❛ ♠❛t❡♠át✐❝❛ ❡ ❢❛③❡♥❞♦ ❛♣❧✐❝❛çõ❡s t❛♥t♦ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❢ís✐❝❛ q✉❛♥t♦ ♥❛ ❞❡ ♠❛t❡♠át✐❝❛✳

➱ r❡❝♦♠❡♥❞á✈❡❧ q✉❡ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❛❜♦r❞❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❱❡t♦r t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦ ✭❝♦❧❡çã♦ ❞❡ s❡❣♠❡♥t♦s ❡q✉✐♣♦❧❡♥t❡s✮ q✉❛♥t♦ ❛❧❣é❜r✐❝♦ ✭❝❛✲ r❛❝t❡r✐③❛❞♦ ♣♦r ❝♦♦r❞❡♥❛❞❛s✮✳ ❆ ✐♥❝❧✉sã♦ ❞❛ ♥♦çã♦ ❞❡ ❱❡t♦r❡s ♥❛ ❛✉❧❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ❝♦rr✐❣✐r✐❛ ✉♠ ❡rr♦ ❜ás✐❝♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣❡❧♦ ❢❛t♦ ❞❡ s❡ tr❛t❛r ❞❡ ✉♠ tó♣✐❝♦ ♠❛t❡♠át✐❝♦ ❞❡ ✉♠ ❛❧t♦ ❣r❛✉ ❞❡ r❡❧❡✈â♥❝✐❛✱ ♠❛s q✉❡ é tr❛t❛❞♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❛♣❡♥❛s ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❋ís✐❝❛✳ ➱ ✐♥t❡r❡ss❛♥t❡ t❛♠❜é♠ ✉t✐❧✐③❛r ♣r♦❜❧❡♠❛s ❞❛ ❆♥t✐❣✉✐❞❛❞❡ r❡s♦❧✈✐❞♦s ❛tr❛✈és ❞❡ ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s✳ ❈♦♠♦ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ♣❡sq✉✐s❛✱ ❢♦✐ ❛❞♦t❛❞❛ ❛ ♣❡sq✉✐s❛ ❜✐❜❧✐♦❣rá✲ ✜❝❛✱ ♣❛r❛ ♣r♦❞✉③✐r ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❛❜♦r❞❛❣❡♠ ❞♦s ❝♦♥t❡ú❞♦s s♦❜ ✉♠❛ ót✐❝❛ ❣❡♦♠étr✐❝❛ ❡ ❛❧❣é❜r✐❝❛✱ ❜✉s❝❛♥❞♦ ♦ ❡q✉✐❧í❜r✐♦ ❡♥tr❡ ❝♦♥❝❡✐t♦s ❛❜♦r❞♦s ❡ ❛♣❧✐❝❛çõ❡s✳

◆♦ ❈❛♣ít✉❧♦ ✷ ✐♥tr♦❞✉③✐♠♦s ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s ❣❡♦♠étr✐❝❛♠❡♥t❡ ❞❡ ♠♦❞♦ ❝♦♥s✲ tr✉t✐✈♦ ❡ ♣❡❧❛ ✈✐s✉❛❧✐çã♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠❡ç❛♥❞♦ ♣❡❧❛ t❡♦r✐❛ ✈❡t♦r✐❛❧✱ ❝✐t❛♠♦s t❛♠❜é♠ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ✭✈❡t♦r❡s ♥❛ s✉❛ ❡①ê♥❝✐❛✮✱ ❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛✱ ❡ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❣❡♦♠étr✐❝❛s ❞♦s ✈❡t♦r❡s✳

❖ ❈❛♣ít✉❧♦ ✸ é ❞❡❞✐❝❛❞♦ ❛♦ ❡♥❢♦q✉❡ ❞❡ ♦♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s ✭s♦♠❛✱ ♠✉❧t✐♣❧✐✲ ❝❛çã♦✱ s❡♠♣r❡ ❜✉s❝❛♥❞♦ r❡❧❛❝✐♦♥❛r ❛ ❞❡✜♥✐çã♦ ❝♦♠ ✉♠ ❡①❡♠♣❧♦ ❣❡♦♠étr✐❝♦ r❡❧❛❝✐♦♥❛❞♦ à ♠❛t❡♠át✐❝❛✳

◆♦s ❈❛♣ít✉❧♦s ✹ ❡ ✺ ❞❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çõ❡s ❞❛ t❡♦r✐❛ ❡st✉❞❛❞❛ ♥♦s ❝❛♣ít✉❧♦s ✷ ❡ ✸✱ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ❋ís✐❝❛ ❡ ▼❛t❡♠át✐❝❛✳

◆♦ ❝❛♣ít✉❧♦ ✻ ❢❛③❡♠♦s ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❡ ❱❡t♦r❡s ♥♦ R3✱ ❞❛♥❞♦ ✉♠

❡①❡♠♣❧♦ ♣❛r❛ ❝❛❞❛ ❞❡✜♥✐çã♦ ❡ ♥♦ ✜♠ ❜✉s❝❛♥❞♦ ❛♣❧✐❝❛çõ❡s ❞❛ á❧❣❡❜r❛ ✈❡t♦r✐❛❧ ♣❛r❛ s✐t✉❛✲ çõ❡s q✉❡ ♦ ❛❧✉♥♦ ❡♥❝♦♥tr❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♠❛s ♥ã♦ sã♦ ❛❜♦r❞❛❞❛s ✉s✉❛❧♠❡♥t❡ ♣❡❧♦ s❡✉ ♣r♦❢❡ss♦r✳

❈❛♣ít✉❧♦ ✷

■♥tr♦❞✉çã♦ ❛ ➪❧❣❡❜r❛ ❱❡t♦r✐❛❧

❆ á❧❣❜❡r❛ ✈❡t♦r✐❛❧ é ❛♣r❡s❡♥t❛❞❛ ♠♦str❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ♦♣❡r❛❝✐♦♥❛✐s ❞♦s ✈❡t♦r❡s q✉❡ sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳ P❛r❛ ❡st❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ ✉s❛❞❛s ❛s r❡❢❡r❡♥❝✐❛s ❬✷❪✱ ❬✼❪ ❡ ❬✶✵❪✳

❊①✐st❡♠ três ♠❛♥❡✐r❛s ❞❡ s❡ ✐♥tr♦❞✉③✐r ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s✿

• ●❡♦♠❡tr✐❝❛♠❡♥t❡✿ ♦s ✈❡t♦r❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦s

✭s❡t❛s✮ ❡ ❛s ♦♣❡r❛çõ❡s ❝♦♠ ❡❧❡s sã♦ ❞❡✜♥✐❞❛s ❣❡♦♠❡tr✐❝❛♠❡♥t❡❀

• ❆♥❛❧✐t✐❝❛♠❡♥t❡✿ ♦s ✈❡t♦r❡s ❡ ❝♦rr❡s♣♦♥❞❡♥t❡s ♦♣❡r❛çõ❡s sã♦ ❞❡s❝r✐t♦s ❡♠ t❡r♠♦s

❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❝❤❛♠❛❞♦s ❝♦♠♣♦♥❡♥t❡s ✈❡t♦r✐❛✐s✳ ❆ ❞❡s❝r✐çã♦ ❛♥❛❧ít✐❝❛ r❡s✉❧t❛ ♥❛t✉r❛❧♠❡♥t❡ ❞❛ ❞❡s❝r✐çã♦ ❣❡♦♠étr✐❝❛✱ ❞❡s❞❡ q✉❡ s❡❥❛ ✐♥tr♦❞✉③✐❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s❀

• ❆①✐♦♠❛t✐❝❛♠❡♥t❡✿ ♥ã♦ s❡ ❢❛③ q✉❛❧q✉❡r t❡♥t❛t✐✈❛ ♣❛r❛ s❡ ❞❡s❝r❡✈❡r ✉♠ ✈❡t♦r ♦✉

❛s ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s ❝♦♠ ✈❡t♦r❡s✳ ◆❡st❡ ❝❛s♦✱ ✈❡t♦r❡s ❡ ♦♣❡r❛çõ❡s ✈❡t♦r✐❛✐s sã♦ ❝♦♥s✐❞❡r❛❞♦s ❝♦♥❝❡✐t♦s ♥ã♦ ❞❡✜♥✐❞♦s✱ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦s q✉❛✐s s❡ s❛❜❡ ❛♣❡♥❛s q✉❡ ❡❧❡s s❛t✐s❢❛③❡♠ ❝❡rt♦ ❝♦♥❥✉♥t♦ ❞❡ ❛①✐♦♠❛s✳ ❚❛❧ s✐st❡♠❛ ❛❧❣é❜r✐❝♦✱ ❝♦♠ ❛①✐♦♠❛s ❛♣r♦♣r✐✲ ❛❞♦s✱ ❝❤❛♠❛✲s❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❊♠ t♦❞♦s ♦s r❛♠♦s ❞❛ ▼❛t❡♠át✐❝❛ s❡ ❡♥❝♦♥tr❛♠ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳

◆❡st❛ s❡çã♦✱ ✐♥tr♦❞✉③✐♠♦s ♦ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❞❡ ♠♦❞♦ ❝♦♥s✲ tr✉t✐✈♦ ❡ ♣❡❧❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦ ♠❡s♠♦✱ ❞❡♥tr♦ ❞♦ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❊♠ s❡❣✉✐❞❛✱ ✉t✐❧✐③❛♠♦s ♦ ♠ét♦❞♦ ❛♥❛❧ít✐❝♦ ❡ ❣❡♦♠étr✐❝♦ ♣❛r❛ ✐♥tr♦❞✉③✐r ♦✉tr♦s ❝♦♥❝❡✐t♦s ❡ ♦♣❡r❛çõ❡s✳

✷✳✶ ●r❛♥❞❡③❛s ❊s❝❛❧❛r❡s ❡ ❱❡t♦r✐❛✐s

❈❡rt❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❝♦♠♦ ♠❛ss❛ ♦✉ ❛ t❡♠♣❡r❛t✉r❛ ❛❜s♦❧✉t❛ ♣♦❞❡♠ s❡r ❞❡✲ ✜♥✐❞❛s ♣♦r ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❞❡s❞❡ q✉❡ ❛❝♦♠♣❛♥❤❛❞❛s ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❛❞❡q✉❛❞❛

✉♠ ❡①❡♠♣❧♦ ❝❧❛r♦ é ❛ ❚❡♠♣❡r❛t✉r❛✱ ♣♦✐s é ❞❡✈✐❞❛♠❡♥t❡ ❞❡✜♥✐❞❛ ♣♦r ✉♠ ♥ú♠❡r♦ ❡ s✉❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ❡①❡♠♣❧♦ 30◦_C✱ é ❞❡✜♥✐❞♦ ♣❡❧♦ ♥ú♠❡r♦ ₃₀ ❡ ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ é

♦ (◦_C)✱ ❝♦♠♦ ♥❡ss❡ ❡①❡♠♣❧♦✳ ❊st❛s ❣r❛♥❞❡③❛s sã♦ ❝❤❛♠❛❞❛s ❡s❝❛❧❛r❡s✳ ❊①✐st❡♠✱ ♥♦ ❡♥✲

t❛♥t♦✱ ♦✉tr❛s ❣r❛♥❞❡③❛s ❢ís✐❝❛s q✉❡ tê♠ ✐♥t❡♥s✐♥❞❛❞❡✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✱ sã♦ ❛s ❣r❛♥❞❡③❛s ✈❡t♦r✐❛✐s❀ ❛ ✐♥t❡s✐❞❛❞❡ ♣♦❞❡ ❛✉♠❡♥t❛r ♦✉ ❞✐♠✐♥✉✐r✱ ❡ ❛ ❞✐r❡çã♦ ❡ ♦ s❡♥t✐❞♦ ♣♦❞❡r♠ s❡ ✐♥✈❡rt❡r✳ ❊st❛s q✉❛♥t✐❞❛❞❡s ♣♦❞❡♠ s❡r ❛❞✐❝✐♦♥❛❞♦s ❞❡ ❢♦r♠❛ q✉❡ ❛ s❡ ❝♦♥s✐❞❡r❛r ❞✐r❡çã♦✱ s❡♥t✐❞♦ ❡ ✐♥t❡♥s✐❞❛❞❡ ✭♦✉ ♠ó❞✉❧♦✮✳ ❋♦rç❛ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ q✉❡ ❛t✉❛ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ s❡♥t✐❞♦ ❡ ❞✐r❡çã♦ ❝♦♠ ❛❧❣✉♠❛ ✐♥t❡♥s✐❞❛❞❡ q✉❡ ♠❡❞✐♠♦s ❡♠ ◆❡✇t♦♥s✭◆✮ ✳ ◗✉❛♥❞♦ ❞✉❛s ❢♦rç❛s ❛t✉❛♠ s♦❜r❡ ✉♠ ♠❡s♠♦ ♦❜❥❡t♦✱ ❛ s♦♠❛ ❞❛s ❢♦rç❛s ❞❡♣❡♥❞❡ t❛♥t♦ ❞❛ ❞✐r❡çã♦✱ s❡♥t✐❞♦ ❡ ✐♥t❡♥s✐❞❛❞❡ ❞❛s ❞✉❛s ❢♦rç❛s✳ P♦s✐çã♦✱ ❞❡s❧♦❝❛♠❡♥t♦✱ ✈❡❧♦❝✐❞❛❞❡✱ ❛❝❡❧❡r❛✲ çã♦✱ ❢♦rç❛✱ ✐♠♣✉❧s♦ ❡ t♦rq✉❡ sã♦ t♦❞❛s ❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s q✉❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣♦r ✈❡t♦r❡s✳ ❱❛♠♦s ❝♦♠❡ç❛r ♣♦r ❞❡✜♥✐r ♣r❡❝✐s❛♠❡♥t❡ ♦ q✉❡ ❡♥t❡♥❞❡♠♦s ♣♦r ✉♠ ✈❡t♦r✳

✷✳✷ ❙❡❣♠❡♥t♦s ❞❡ ❘❡t❛

●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ✈❡t♦r❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r s❡❣♠❡♥t♦s ✭❞❡ r❡t❛s✮ ♦r✐❡♥t❛❞♦s ✭s❡❣♠❡♥t♦s ❞❡ r❡t❛s ❝♦♠ ✉♠ s❡♥t✐❞♦ ❞❡ ♣❡r❝✉rs♦✮ ♥♦ ♣❧❛♥♦ ♦✉ ♥♦ ❡s♣❛ç♦✳ ❯♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ✉♠❛ ♣❛rt❡ ❞❡ ✉♠❛ r❡t❛ q✉❡ ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ✉♠ ♣♦♥t♦ ✜♥❛❧✱ ❝❤❛♠❛❞♦s ❞❡ ❡①tr❡♠♦✳ ◆❛ ✜❣✉r❛ 2.1 t❡♠♦s ✉♠❛ r❡t❛ r✱ ❡ ❛ ♣❛rt❡ ❝♦♠♣r❡❡♥❞✐❞❛

❡♥tr❡ ♦s ♣♦♥t♦s ❆ ❡ ❇ é ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✳

❋✐❣✉r❛ ✷✳✶✿ ❙❡❣✉✐♠❡♥t♦ AB ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

◆♦t❡ q✉❡ ♦ s❡❣♠❡♥t♦ AB =BA ✳ ❊ ♦ s❡❣♠❡♥t♦AA s❡rá ❝♦♥s✐❞❡r❛❞♦ s❡❣♠❡♥t♦ ♥✉❧♦✳

❯♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ −→AB é ❞❡✜♥✐❞♦ ♣♦r ✉♠ s❡❣♠❡♥t♦ AB ♠❛✐s ❛ ❡s❝♦❧❤❛ ❞❡ ✉♠ ❞♦s s❡✉s ❡①tr❡♠♦s ❝♦♠♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ♦ ♦✉tr♦ ❝♦♠♦ ♣♦♥t♦ ✜♥❛❧✳ ❖❜s❡r✈❡ ♥❛ ❋✐❣✉r❛

2.2 q✉❡ ❛ ♦r✐❡♥t❛çã♦ ❞♦s s❡❣✉✐♠❡♥t♦s é ❞❛❞❛ ❛ ♣❛rt✐r ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ♣♦♥t♦

✜♥❛❧✳

✭❛✮ ❙❡❣♠❡♥t♦ ❖r✐❡♥t❛❞♦AB ✭❜✮ ❙❡❣♠❡♥t♦ ❖r✐❡♥t❛❞♦BA

❋✐❣✉r❛ ✷✳✷✿ ❙❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❊①❡♠♣❧♦ ✷✳✶✳ ❖❜s❡r✈❛♥❞♦ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦✭❋✐❣✉r❛ 2.3✮✱ ♥♦t❛♠♦s q✉❡ ❡❧❡ t❡♠ ✈❛r✐♦s

s❡❣✉✐♠❡♥t♦s ♦r✐❡♥t❛❞♦s r❡♣r❡s❡♥t❛❞♦s ♥❛ ✜❣✉r❛✳ ❖ s❡❣♠❡♥t♦ −→AB t❡♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ A ❡ ♣♦♥t♦ ✜♥❛❧B✳ ❱❛❧❡ ❞❡st❛❝❛r q✉❡−→AB é ❞✐❢❡r❡♥t❡ ❞❡ −→BA

❋✐❣✉r❛ ✷✳✸✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ABCDEF GH ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

✷✳✸ ◆♦r♠❛✱ ❉✐r❡çã♦ ❡ ❙❡♥t✐❞♦

P❛r❛ ❡❢❡✐t♦ ❞❛ ❞❡✜♥✐çã♦ ❡ ❡st✉❞♦ ❞❡ ✈❡t♦r❡s✱ ♣r❡❝✐s❛♠♦s ❝♦♠♣❛r❛r ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❛ ✉♠ ♦✉tr♦✱ ♦❜s❡r✈❛♥❞♦ ❛s três s❡❣✉✐♥t❡s ❝❛r❛❝t❡ríst✐❝❛s✿

• ◆♦r♠❛✿ é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ −→AB ✱ ❞❡♥♦t❛❞♦ ♣♦r_k−→AB_k✳

• ❉✐r❡çã♦✿ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s −→AB ❡−−→CD t❡rã♦ ♠❡s♠❛ ❞✐r❡çã♦ s❡ ❛s r❡t❛s q✉❡ ♦s ❝♦♥té♠ sã♦ ❝♦✐♥❝✐❞❡♥t❡s ♦✉ ♣❛r❛❧❡❧❛s✳

• ❙❡♥t✐❞♦✿ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s −→AB ❡ −−→CD q✉❡ t✐✈❡r❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ ♥ã♦ ❢♦r❡♠ ❝♦❧✐♥❡❛r❡s✱ tê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦ q✉❛♥❞♦ −→AC _∩ −−→BD ❂ _{} ✱ ❝❛s♦ ❝♦♥trár✐♦ tê♠ s❡♥t✐❞♦s♦♣♦st♦s✳ ❖s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s −→AB ❡ −−→CD ❝♦❧✐♥❡❛r❡s t❡rã♦ ♦ ♠❡s♠♦

s❡♥t✐❞♦✱ q✉❛♥❞♦ ✉♠ ♦✉tr♦ s❡❣♠❡♥t♦ ❛✉①✐❧✐❛r−−→A′_B′ ♥ã♦ ❝♦❧✐♥❡❛r ❝♦♠−−→_CD ❡ ♥♦ ♠❡s♠♦

s❡♥t✐❞♦ ❞❡ −→AB ✱ s❛t✐s❢❛③❡r −→AC _∩−−→BD ❂_{}✳

❊①❡♠♣❧♦ ✷✳✷✳ ❆✐♥❞❛ ❝♦♠ r❡❧❛çã♦ ❛ ✜❣✉r❛2.3✱ ♣♦❞❡♠♦s ❧✐st❛r ✈ár✐♦s ❡①❡♠♣❧♦s ❞❡ ◆♦r♠❛✱

❞✐r❡çã♦ ❡ s❡♥t✐❞♦

• {AD, CB, HE, F G_}é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s q✉❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♥♦r♠❛✳

• {AB, EF , DC, HG_} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s q✉❡ ♣♦ss✉❡♠ ♠❡s♠♦ s❡♥t✐❞♦✳

• {AB, BA, EH, HE_}é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s q✉❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✳

✷✳✹ ❙❡❣✉✐♠❡♥t♦s ❊q✉✐♣♦❧❡♥t❡s

❉❡✜♥✐çã♦ ✷✳✶ ✭❙❡❣✉✐♠❡♥t♦s ❡q✉✐♣♦❧❡♥t❡s✮✳ ❉♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s −→AB ❡ −−→CD✱ ♥ã♦ ♥✉❧♦s✱ sã♦ ❡qü✐♣♦❧❡♥t❡s s❡ ♦s s❡❣♠❡♥t♦s t✐✈❡r❡♠ ❛ ♠❡s♠❛ ♥♦r♠❛✱ ♠❡s♠❛ ❞✐r❡çã♦ ❡ ♠❡s♠♦ s❡♥t✐❞♦✱ ❡ r❡♣r❡s❡♥t❛r❡♠♦s ❡ss❛ r❡❧❛çã♦ ❝♦♠−→AB _≡−−→CD✳

❋✐❣✉r❛ ✷✳✹✿ ❙❡❣✉✐♠❡♥t♦ ❡q✉✐♣♦❧❡♥t❡s ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

Pr♦♣r✐❡❞❛❞❡s ❞❡ ❙❡❣✉✐♠❡♥t♦s ❊q✉✐♣♦❧❡♥t❡s

❉❛❞♦s três s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s q✉❛✐sq✉❡r −→AB ✱ −−→CD ❡ −→EF ✱ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✲ ♣♦❧❡♥❝✐❛ ❣♦③❛ ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(i) r❡✢❡①✐✈❛✿−→AB _≡−→AB❀

(ii) s✐♠étr✐❝❛✿ s❡ −→AB _≡−−→CD✱ ❡♥tã♦ −−→CD _≡−→AB❀

(iii) tr❛♥s✐t✐✈❛✿ s❡ −→AB _≡−−→CD ❡ CD−−→_≡−→EF✱ ❡♥tã♦ −→AB _≡−→EF✿

❖❜s❡r✈❛çõ❡s✿

• ❚♦❞♦s ♦s s❡❣♠❡♥t♦s ♥✉❧♦s sã♦ ❡qü✐♣♦❧❡♥t❡s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ −→AA_≡−−→BB ✳

• ❚♦❞❛ r❡❧❛çã♦ q✉❡ é r❡✢❡①✐✈❛✱ s✐♠étr✐❝❛ ❡ tr❛♥s✐t✐✈❛ é ❝❤❛♠❛❞❛ ❞❡ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛✲

❧ê♥❝✐❛✱ ❧♦❣♦ ❛ ❡qü✐♣♦❧ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳

• ◆♦t❡ q✉❡✱ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡qü✐♣♦❧ê♥❝✐❛✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ❡♠ q✉❛❧q✉❡r

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

❊①❡♠♣❧♦ ✷✳✸✳ ❖❜s❡r✈❛♥❞♦ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦✭❋✐❣✉r❛ 2.5✮✱ ♥♦t❛✲s❡ ✈ár✐♦s ❝♦♥❥✉♥t♦s ❞❡

s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✳ ♥♦t❛♠♦s q✉❡ ❡❧❡ t❡♠ ✈❛r✐♦s s❡❣✉✐♠❡♥t♦s ♦r✐❡♥t❛❞♦s r❡♣r❡s❡♥t❛❞♦s ♥❛ ✜❣✉r❛ ✷✳✺✳

• {−→AB,−−→DC,−→EF ,−−→HG_} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✳

• {−−→AD,−−→BC,−−→EH,−→F G_} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✳

• {−→AE,−−→BF ,−→CG,−−→EH_} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✳

❋✐❣✉r❛ ✷✳✺✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ABCDEF GH ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

✷✳✺ ❱❡t♦r❡s

■♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦✱ ♦ q✉❡ ♣❡r♠✐t❡ ✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ♣❛r❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ♠♦❞♦ ✐♥t✉✐tí✈♦ ❞♦s ✈❡t♦r❡s✳ P♦r ✐ss♦✱ ✈❛♠♦s ♥♦s r❡str✐♥❣✐r ❛♦ ♣❧❛♥♦ ❡ ♠❛✐s t❛r❞❡✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛r♠♦s ✈❡t♦r❡s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛❧❣é❜r✐❝♦✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t✐rá ❡st✉❞❛r ✈❡t♦r❡s ❡♠ ❡s♣❛ç♦s ❞❡ tr✐❞✐♠❡♥s✐♦♥❛✐s✳

◆♦t❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❡ ✉♠ ✈❡t♦r✳ ❯♠ s❡❣♠❡♥t♦ ♦r✐❡♥✲ t❛❞♦ é ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞✐r❡❝✐♦♥❛❞♦✱ ♦ q✉❛❧ é ✜①❛❞♦ ❡ t❡♠ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ✜♥❛❧

✜①❛❞♦s✳ ❯♠ ❱❡t♦r é ✉♠ r❡♣r❡s❡♥t❛♥t❡ ❞❛ ❝❧❛ss❡ ❞♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡qü✐♣♦❧❡♥t❡s ❛ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❞❛❞♦ q✉❛❧q✉❡r✱ ♦✉ s❡❥❛✱ ♦ ✈❡t♦r ♥ã♦ é ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ✭❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✮ ❡s♣❡❝í✜❝♦✱ ♠❛s ✉♠ r❡♣r❡s❡♥t❛♥t❡ ❞♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s q✉❡ t❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✱ ♠❡s♠♦ s❡♥t✐❞♦ ❡ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❛❞♦✳

❋✐❣✉r❛ ✷✳✻✿ ❱❡t♦r~v ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❖❜s❡r✈❛çã♦✿

❖ ✈❡t♦r ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦AB s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r−→AB✱ ♦✉ ♣♦r ✉♠❛ ❧❡tr❛ ♠✐♥ús❝✉❧❛~v✳

❖s ✈❡t♦r❡s ♣♦ss✉❡♠ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❡s♣❡❝✐❛✐s✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞♦ ♠ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r✱ q✉❡ é ❛ ♠❡❞✐❞❛ q✉❡ ♦❜t❡♠♦s q✉❛♥❞♦ ❝♦♠♣❛r❛♠♦s ✉♠ ✈❡t♦r ❝♦♠ ♦✉tr♦ ❞❡ ♠❡s♠❛ ❡s♣é❝✐❡✱ ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♥✐❞❛❞❡✳ P♦r ❡①❡♠♣❧♦✿ ♦ ♠ó❞✉❧♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✉♠ ❝❛rr♦ ❡♠ ❝❡rt♦ ✐♥st❛♥t❡ é ❞❡50km/h✱ s❡ ♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡ ❛❞♦t❛❞♦ ❝♦♠♦ ✉♥✐tár✐♦ ❡st✐✈❡r ❝♦♥t✐❞♦ 50 ✈❡③❡s ♥♦ ✈❡t♦r ❝♦♥s✐❞❡r❛❞♦ q✉❡ ♣♦ss✉✐ ✈ár✐❛s ❛♣❧✐❝❛❜✐❧✐❞❛❞❡s t❛♥t♦ ❡♠ ❋ís✐❝❛

q✉❛♥t♦ ❡♠ ▼❛t❡♠át✐❝❛✳ ◆❛ ♠❛t❡♠át✐❝❛ ♦ ♠ó❞✉❧♦ ❞♦ ✈❡t♦r ✐♥❞✐❝❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ q✉❡ ♦ r❡♣r❡s❡♥t❛ ❡♥q✉❛♥t♦ q✉❡ ♥❛ ❋ís✐❝❛ ♣♦❞❡ r❡♣r❡s❡♥t❛r ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ✉♠❛ ❢♦rç❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ ✈❡t♦r✳ ❯♠ ✈❡t♦r ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✶ é ❝❤❛♠❛❞♦ ✈❡t♦r ✉♥✐tár✐♦✳ P❛r❛ ✐♥❞✐❝❛r♠♦s ♦ ♠ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r✱ ✉s❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦ k~v_k ❡ ♦ ♠ó❞✉❧♦ ❞♦ ✈❡t♦r~v é ✐♥❞✐❝❛❞♦ ♣♦r _k~v_k=√~v_·~v✳

❉✐③❡♠♦s q✉❡ ❞♦✐s ✈❡t♦r❡s ♦✉ ♠❛✐s sã♦ ✐❣✉❛✐s ♦✉ ❡q✉✐♣♦❧❡♥t❡s s❡ s❡✉s ♠ó❞✉❧♦s t❛♠❜é♠ ❢♦r❡♠ ✐❣✉❛✐s✱ s❡ s✉❛s ❞✐r❡çõ❡s ❢♦r❡♠ ✐❣✉❛✐s ❡ s❡ ♣♦ss✉ír❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✳ P♦ré♠✱ q✉❛♥❞♦ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡ é ❞✐❢❡r❡♥t❡✱ ❞✐③❡♠♦s q✉❡ ♦s ✈❡t♦r❡s sã♦ ❞✐❢❡r❡♥t❡s✳ ❈❤❛♠❛♠♦s ❞❡ ✈❡t♦r ♦♣♦st♦ ❞❡ ✉♠ ✈❡t♦r~v ❂−→AB✱ ♦ ✈❡t♦r−→BA ✐♥❞✐❝❛❞♦ ♣♦r₋~v ♦✉₋−→AB✱ q✉❡ ♣♦ss✉✐ ♦ ♠❡s♠♦ ♠ó❞✉❧♦✱ ♠❡s♠❛ ❞✐r❡çã♦✱ ♣♦ré♠ s❡✉ s❡♥t✐❞♦ é ♦♣♦st♦ ❛♦ ❞❡~v✳

❉❡ ♠♦❞♦ ❣❡r❛❧✱ ❝♦♥❝❡✐t♦s ❡♥✈♦❧✈❡♥❞♦ ✈❡t♦r❡s sã♦ ❞❡✜♥✐❞♦s ✉t✐❧✐③❛♥❞♦ s❡✉s r❡♣r❡✲ s❡♥t❛♥t❡s✳ ❋✐♥❛❧✐③❛♥❞♦ ❡st❛ s❡çã♦✱ ❞♦✐s ✈❡t♦r❡s~v❡~usã♦ ❞✐t♦s ♦rt♦❣♦♥❛✐s✱ s❡ ❛♦ ❡s❝♦❧❤❡r♠♦s ❞♦✐s r❡♣r❡s❡♥t❛♥t❡s ♣❛r❛ ❡ss❡s ✈❡t♦r❡s q✉❡ ✐♥✐❝✐❛♠ ♥♦ ♠❡s♠♦ ♣♦♥t♦✱−→AB ❡ −→AC ❡ss❡s s❡❣✲

♠❡♥t♦s ❢♦r❡♠ ♦rt♦❣♦♥❛✐s✱ ♦✉ s❡❥❛✱ s❡ ♦ â♥❣✉❧♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ❡ss❡s s❡❣♠❡♥t♦s ❢♦r ✉♠ â♥❣✉❧♦ r❡t♦✳

❋✐❣✉r❛ ✷✳✼✿ ❱❡t♦r❡s ♦rt♦❣♦♥❛✐s ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❈❛♣ít✉❧♦ ✸

❖♣❡r❛çõ❡s ❝♦♠ ❱❡t♦r❡s

❖♣❡r❛r ❝♦♠ ✈❡t♦r❡s é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❛♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r❡s✱ ♣♦✐s ♦ ♠❡✲ ❝❛♥✐s♠♦ é ❞✐❢❡r❡♥t❡ ❞❛ ♦♣❡r❛çã♦ ❝♦♠ ♥ú♠❡r♦s r❡❛✐s✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❡♥✈♦❧✈❡ ❛♣❡♥❛s ✈❛❧♦r❡s ♥✉♠ér✐❝♦s✱ ♠❛s t❛♠❜é♠ ♦r✐❡♥t❛çõ❡s ❡s♣❛❝✐❛✐s✳ P♦rt❛♥t♦✱ ❛s r❡❣r❛s ♣❛r❛ ❛ á❧❣❡❜r❛ ❞❡ ✈❡t♦r❡s sã♦ ❞✐❢❡r❡♥t❡s ❞❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ á❧❣❡❜r❛ ❞♦s ♥ú♠❡r♦s✳ ◆❡ss❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛s ♦♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s✳ P❛r❛ t❛♥t♦ ✉s❛♠♦s ❛s s❡❣✉✐♥t❡s r❡❢❡r❡♥❝✐❛s ❬✷❪✱ ❬✽❪✱ ❬✶✵❪ ❡ ❬✶✶❪✳

✸✳✶ ❙♦♠❛ ❞❡ ❱❡t♦r❡s

❆ s♦♠❛ ❞❡ ❞♦✐s ✈❡t♦r❡s~u❡~vq✉❛✐sq✉❡r✱ é ♦❜t✐❞❛ ❣r❛✜❝❛♠❡♥t❡✱ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛ ✭✈❡r ✜❣✉r❛ ✸✳✶✮✿

• ❊s❝♦❧❤❡✲s❡ ✉♠ ♣♦♥t♦ A q✉❛❧q✉❡r❀

• ❉♦ ♣♦♥t♦ A ❝♦♥str✉❛ ✉♠ ♦✉tr♦ r❡♣r❡s❡♥t❛♥t❡ ♣❛r❛ ♦ ✈❡t♦r ~u✱ ♦✉ s❡❥❛✱ ~u=−→AB❀

• ❉♦ ♣♦♥t♦ ❇ ❝♦♥str✉❛ ✉♠ ♦✉tr♦ r❡♣r❡s❡♥t❛♥t❡ ♣❛r❛ ♦ ✈❡t♦r~v✱ ♦✉ s❡❥❛✱~v =−−→BC❀ ❖ ✈❡t♦r s♦♠❛ w~ =~u+~v s❡rá r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ✈❡t♦r w~ =AC✳

❋✐❣✉r❛ ✸✳✶✿ ❙♦♠❛ ❞♦s ✈❡t♦r❡s ~u ❡~v

Pr♦♣r✐❡❞❛❞❡s

❉❛❞♦s três ✈❡t♦r❡s~u✱~v ❡w~ q✉❛✐sq✉❡r ❛ s♦♠❛ ❞❡ ✈❡t♦r❡s ❣♦③❛ ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

❬S1 ❪ ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ ~u+~v =~v+~u ❉❛ ✜❣✉r❛ ✷✳✺✱ t❡♠♦s✿

~u+~v =−→AB+−−→BC =AC−→=−−→AD+−−→DC =~v+~u

❬S2 ❪❊❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ✈❡t♦r s♦♠❛✿ ~u+~0 =~0 +~u=~u ❉❛ ✜❣✉r❛ ✸✳✶✱ t❡♠♦s✿

~u+~0 =−→AB +−−→BB=AB−→=−→AA+−→AB =~0 +~u

❬S3 ❪❉✐str✐❜✉t✐✈✐❞❛❞❡ ❞❛ s♦♠❛~u_×(~v+w~) =~u_×~v+~u_×w~ ❉❛ ✜❣✉r❛ ✸✳✶✱ t❡♠♦s✿

~u+ (₋~u) =−→AB +−→BA =−→AA =~0

❬S4 ❪❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ (~u+~v) +w~ =~u+ (~v+w~)

❉❛ ✜❣✉r❛ ✸✳✷✱ t❡♠♦s✿

(~u+~v)+w~ = (−→AB+−−→BC)+−−→CD =−→AC+−−→CD =−−→AD=−→AB+−−→BD=−→AB+(−−→BC+−−→CD) =

~u+ (~v+w~)

❋✐❣✉r❛ ✸✳✷✿ ❙♦♠❛ ❞♦s ✈❡t♦r❡s ~u✱~v ❡ w~ ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

✸✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❱❡t♦r❡s ♣♦r ✉♠ ❊s❝❛❧❛r

❉❡✜♥✐çã♦ ✸✳✶ ✭▼✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❊s❝❛❧❛r✮✳ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ✈❡t♦r~v ♣♦rα é ✉♠ ♥♦✈♦ ✈❡t♦r r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ α~v✱ q✉❡ t❡♠ ♠❡s♠❛ ❞✐r❡çã♦ ❞♦ ✈❡t♦r ~v✱ ♥♦r♠❛ ✐❣✉❛❧ ❛

|α_|k~v_k✱ ♠❡s♠♦ s❡♥t✐❞♦ ❞♦ ✈❡t♦r~v✱ s❡ α >0 ❡✱ s❡ α <0✱ s❡♥t✐❞♦ ♦♣♦st♦✳

✭❛✮ ✭❜✮

❋✐❣✉r❛ ✸✳✸✿ ▼✉❧t✐♣❧✐❝❛çã♦ ❞♦ ✈❡t♦r~v ♣♦r ✭❛✮ α >0✱ ❡ ✭❜✮ ♣♦r α <0✳

❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❖❜s❡r✈❛çã♦✿

◗✉❛❧q✉❡r ✈❡t♦r ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r α = 0 s❡rá ♦ ✈❡t♦r ♥✉❧♦✱ ♦✉ s❡❥❛✱ 0~v = 0 ❡ q✉❛❧q✉❡r

❡s❝❛❧❛rα r❡❛❧ ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧♦ ✈❡t♦r ♥✉❧♦ s❡rá ♦ ✈❡t♦r ♥✉❧♦✱ ✐st♦ é α~0 = 0✳

Pr♦♣r✐❡❞❛❞❡s

❉❛❞♦s ♦s ✈❡t♦r❡s~u ❡~v q✉❛s✐q✉❡r ❡ ♦s ♥ú♠❡r♦s r❡❛✐s α ❡ β✱ t❡♠♦s✿

❬M1 ❪ Pr♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ❞♦ ❡s❝❛❧❛r ❡♠ r❡❧❛çã♦ à s♦♠❛ ❞❡ ✈❡t♦r❡s✿

α(~u+~v) =α~u+α~v❀

❬M2 ❪ Pr♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ❞♦ ✈❡t♦r ❡♠ r❡❧❛çã♦ à s♦♠❛ ❞♦s ❡s❝❛❧❛r❡s✿

(α+β)~v =α~v+β~v

❬M3 ❪ ❊❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✿

1_·~u =~u

❬M4 ❪α(β~u) = (αβ)~u

❖❜s❡r✈❛çã♦✿

❯♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ♦♥❞❡ sã♦ ❞❡✜♥✐❞❛s ❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❡ q✉❡ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ s♦♠❛ ❙✶✱ ❙✷✱ ❙✸✱ ❙✹ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ▼✶✱ ▼✷✱ ▼✸ ❡ ▼✹ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ss❡ ❝♦♥❥✉♥t♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈❡t♦r❡s✳

❊①❡♠♣❧♦ ✸✳✶✳ ❖❜s❡r✈❡ ♥❛ ✜❣✉r❛ 3.4 ♦ ✈❡t♦r ~v✱ ❞❡♣♦✐s ♦ ♠❡s♠♦ ✈❡t♦r ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ❡s❝❛❧❛r❡s2~v ❡₋4~v✳

❋✐❣✉r❛ ✸✳✹✿ ❱❡t♦r❡s~v✱ 2~v ❡₋4~v ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

✸✳✸ ❈♦♠❜✐♥❛çã♦ ▲✐♥❡❛r

❉❡✜♥✐çã♦ ✸✳✷ ✭❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r✮✳ ❯♠ ✈❡t♦r ~v é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ✈❡t♦r❡s ~v1, ~v2,· · · , ~vn s❡ ❡①✐st✐r❡♠ ♥ú♠❡r♦s r❡❛✐sα1, α2,· · ·αn✱ t❛✐s q✉❡ ♦ ✈❡t♦r~v ♣♦ss❛ s❡r ❡s❝r✐t♦

♥❛ ❢♦r♠❛✿ ~v =α1~v1, α2~v2,· · · , αn~vn✳

❖s ♥ú♠❡r♦s α1, α2,· · ·αn sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦❡✜❝✐❡♥t❡s ❞♦ ✈❡t♦r ~v ❡♠ r❡❛❧❛çã♦ ❛♦s

✈❡t♦r❡s~v1, ~v2,· · · , ~vn✳

❋✐❣✉r❛ ✸✳✺✿ ❱❡t♦r w~ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡~v ❡~u ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❊①❡♠♣❧♦ ✸✳✷✳ ❉❛ ✜❣✉r❛ 3.5✱ t❡♠♦s q✉❡ w~ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ~v ❡ ~u✱ ♣♦✐s ~

w= 2~u+ 1~v✳

❖❜s❡r✈❛çã♦✿

❚♦❞♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦ ✈❡t♦r ♥✉❧♦✳

✸✳✹ ❉❡♣❡♥❞ê♥❝✐❛ ❡ ■♥❞❡♣❡♥❞ê♥❝✐❛ ▲✐♥❡❛r

❉❡✜♥✐çã♦ ✸✳✸ ✭❉❡♣❡♥❞ê♥❝✐❛ ❡ ■♥❞❡♣❡♥❞ê♥❝✐❛ ▲✐♥❡❛r✮✳ ❖s ✈❡t♦r❡s ~v1, ~v2,· · ·, ~vn s❡rã♦✱

sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥❞t❡ ✭▲❉✮✱ s❡ q✉❛❧q✉❡r ✉♠ ❞♦s ✈❡t♦r❡s✱ ❢♦r ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♦✉tr♦s n₋1 ✈❡t♦r❡s r❡st❛♥t❡s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❞✐r❡♠♦s q✉❡ sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥✲

❞❡♥t❡s✭▲■✮✳

❖❜s❡r✈❛çõ❡s✿

• ❉♦✐s ✈❡t♦r❡s ~u ❡~v sã♦ ▲❉ s❡ t✐✈❡r❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✱ ♦✉ s❡❥❛✱ ✉♠ ❢♦r ♠ú❧t✐♣❧♦ ❞♦ ♦✉tr♦✿ ~u=α~v❀

• ❚rês ✈❡t♦r❡s ~u✱~v ❡ w~ sã♦ ▲❉✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ sã♦ ♣❛r❛❧❡❧♦s ❛ ✉♠ ♣❧❛♥♦❀

• ◗✉❛tr♦ ✈❡t♦r❡s q✉❛✐sq✉❡r sã♦ s❡♠♣r❡ ▲❉ ♥✉♠ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳

❊①❡♠♣❧♦ ✸✳✸✳ ❉❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ✈❡t♦r❡s~u✱~v ❡ w~✱ t❡♠♦s✿

• ❖s ✈❡t♦r❡s −→AB✱ −→AC ❡ −−→AD sã♦ ▲❉❀

• −→AB ❡−−→DC sã♦ ▲❉❀

• ~u✱~v ❡ w~ sã♦ ▲■✳

❋✐❣✉r❛ ✸✳✻✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s~u✱~v ❡ w~ ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

✸✳✺ ❇❛s❡

❉❡✜♥✐çã♦ ✸✳✹ ✭❇❛s❡✮✳ ❖ ❝♦♥❥✉♥t♦ {~v1, ~v2,· · · , ~v_n} é ✉♠❛ ❜❛s❡ ♣❛r❛ Rn s❡ ~v1, ~v2,· · · , ~v_n

❢♦r❡♠ ✈❡t♦r❡s ▲■ ❞❡Rn✳

❖✉ s❡❥❛✱ ❜❛s❡ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s ▲✐♥❡❛r♠❡♥t❡ ■♥❞❡♣❡♥❞❡♥t❡s q✉❡ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ tr✐❞✐♠❡♥s✐♦♥❛❧✳

❊①❡♠♣❧♦ ✸✳✹✳ ❊①❡♠♣❧♦ ❞❛ ❜❛s❡ ❝❛♥ô♥✐❝❛✱ ♦s ✈❡t♦r❡s~i,~j,~k ♥ã♦ sã♦ ❝♦♣❧❛♥❛r❡s✳ ▲♦❣♦ sã♦ ▲✐♥❡❛r♠❡♥t❡ ■♥❞❡♣❡♥❞❡♥t❡s✱ ❝♦♥s❡q✉❡♥t✐♠❡♥t❡ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♥♦ ❡s♣❛ç♦R3✳

❋✐❣✉r❛ ✸✳✼✿ ❇❛s❡ ❝❛♥ô♥✐❝❛ ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❊①❡♠♣❧♦ ✸✳✺✳ ❊♠ r❡❛❧❛çã♦ ❛ ✜❣✉r❛3.6✱ t❡♠♦s✿

• ❖ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s {~u, ~v, ~w_} é ✉♠❛ ❜❛s❡ ❞♦ R3

✱ ♣♦✐s sã♦ três ✈❡t♦r❡s ✭▲■✮ ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧❀

• ❖ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s {~u, ~w_} ♥ã♦ é ✉♠❛ ❜❛s❡ ❞♦R3✱ ♣♦✐s sã♦ ❛♣❡♥❛s ❞♦✐s ✈❡t♦r❡s✳

✸✳✻ ❇❛s❡ ❖rt♦❣♦♥❛❧ ❡ ❖rt♦♥♦r♠❛❧

❉❡✜♥✐çã♦ ✸✳✺ ✭❇❛s❡ ❖rt♦❣♦♥❛❧ ❡ ❖rt♦♥♦r♠❛❧✮✳ ❯♠❛ ❜❛s❡ {~v1, ~v2,· · · , ~vn} ♣❛r❛ ♦ Rn é

❝❤❛♠❛❞❛ ❞❡ ❜❛s❡ ♦rt♦❣♦♥❛❧ s❡ s❡✉s ✈❡t♦r❡s sã♦ ♦rt♦❣♦♥❛✐s ❞♦✐s ❛ ❞♦✐s✱ ❡ ❞❡ ❜❛s❡ ♦rt♦♥♦r♠❛❧ s❡ ❛❧é♠ ❞❡ s❡r❡♠ ♦rt♦❣♦♥❛✐s✱ s❡✉s ✈❡t♦r❡s sã♦ ✉♥✐tár✐♦s✱ ♦✉ s❡❥❛✱ ❞❡ ♥♦r♠❛ ✐❣✉❛❧ ❛ ✶✳ ❊①❡♠♣❧♦ ✸✳✻✳ ❉❛ ✜❣✉r❛3.6 ❋✐❝❛ ❝❧❛r♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ _{~u✱~v✱ w~_}é ✉♠❛ ❜❛s❡ ❞♦R3✱ ♣♦✐s

s❡✉s ✈❡t♦r❡s sã♦ ♦rt♦❣♦♥❛✐s ❞♦✐s ❛ ❞♦✐s✳

❊①❡♠♣❧♦ ✸✳✼✳ ❉❛ ✜❣✉r❛ 3.6 ❋✐❝❛ ❝❧❛r♦ q✉❡ ♦ ❝♦♥❥✉♥t♦

n _~u

k~u_k, ~v

k~v_k, ~ w

kw~_k

o

é ✉♠❛ ❜❛s❡ ❞♦ R3 ♣♦✐s s❡✉s ✈❡t♦r❡s sã♦ ♦rt♦❣♦♥❛✐s ❞♦✐s ❛ ❞♦✐s ❡ ✉♥✐tár✐♦s✳

✸✳✼ ➶♥❣✉❧♦ ❡♥tr❡ ❱❡t♦r❡s

❉❡✜♥✐çã♦ ✸✳✻ ✭➶♥❣✉❧♦ ❡♥tr❡ ✈❡t♦r❡s✮✳ ❖ â♥❣✉❧♦ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s ~u ❡ ~v ♥ã♦✲♥✉❧♦s é ♦ ♠❡♥♦r â♥❣✉❧♦ θ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s ~u ❡ ~v t❡♥❞♦ ❡❧❡s ♦ ♠❡s♠♦ ♣♦♥t♦ ✐♥✐❝✐❛❧✱ ♦♥❞❡

0_≤θ _≤π✭0_≤900

≤1800

✮✳

❋✐❣✉r❛ ✸✳✽✿ ➶♥❣✉❧♦ ❡♥tr❡ ♦s ✈❡t♦r❡s ~u ❡~v ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

✸✳✽ Pr♦❞✉t♦ ❡♥tr❡ ❱❡t♦r❡s

❖s ♣r♦❞✉t♦s ❡♥tr❡ ✈❡t♦r❡s sã♦ ♦♣❡r❛çõ❡s q✉❡ tr❛③❡♠ ✉♠ ❛♣❡❧♦ ❣❡♦♠étr✐❝♦ ❜❡♠ ✐♥t❡r❡ss❛♥t❡ ❡ q✉❡ s❡rã♦ ♠✉✐t♦ út❡✐s ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ r❡s♦✲ ❧✉çõ❡s ❞❡ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s✱ ♣♦✐s ❡st❡s ♣r♦❞✉t♦s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛s ❣r❛♥❞❡③❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✭♣r♦❞✉t♦ ✐♥t❡r♥♦✮✱ ár❡❛ ✭♣r♦❞✉t♦ ✈❡t♦r✐❛❧✮ ❡ ✈♦❧✉♠❡ ✭♣r♦❞✉t♦ ♠✐st♦✮✱ ❣❡r❛❞♦ ♣♦r ✈❡t♦r❡s ❡♠ ❝❡rt❛s ❝♦♥❞✐çõ❡s✳

✸✳✾ Pr♦❞✉t♦ ■♥t❡r♥♦

❉❡✜♥✐çã♦ ✸✳✼ ✭Pr♦❞✉t♦ ■♥t❡r♥♦✮✳ ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s~u❡~v✱ é ♦ ♥ú♠❡r♦ ~u_·~v ❞❡✜♥✐❞♦ ♣❡❧❛ ❡①♣r❡ssã♦✿

~u_·~v =_k~u_{k · k}~v_{k ·}cosθ

◆♦t❡ q✉❡ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s✉r❣❡ ❞❡ ✉♠❛ r❛③ã♦ tr✐❣♦♥♦♠étr✐❝❛ ❡♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❆❇❈

❈♦♥s✐❞❡r❛♥♦ ♦ ✈❡t♦r ~u ❝♦♠♦ s❡♥❞♦ ✉♠ ✈❡t♦r ✉♥✐tár✐♦✱ ❡ ✉s❛♥❞♦ ❛s r❡❧❛çõ❡s tr✐✲ ❣♦♥♦♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦ ❞❛ ❋✐❣✉r❛ ✸✳✾ ✱ t❡♠♦s✿

k−→AC_k=_k~v_{k ·}cos(θ) = _k~v_kkk~u_·cos(~v, ~u) =_|~u_·~v_|

♦✉ s❡❥❛✱ ♣♦❞❡♠♦s ✈❡r ❡st❡ ♥ú♠❡r♦ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♣r♦❥❡çã♦ ❞♦ ✈❡t♦r~v ❡♠ r❡❧❛çã♦ à ❞✐r❡çã♦ ❞♦ ✈❡t♦r ✉♥✐tár✐♦~u✳

❋✐❣✉r❛ ✸✳✾✿ ❚r✐â♥❣✉❧♦ ❆❇❈ ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

❊①❡♠♣❧♦ ✸✳✽✳ ❙❡♥❞♦ k~u_k= 2 ✱ _k~v_k= 3 ❡1200 ♦ â♥❣✉❧♦ ❡♥tr❡

~u ❡~v✱ ❈❛❧❝✉❧❡ ~u_·~v✳

P❡❧❛ ❞❡✜♥✐çã♦ t❡♠♦s✿

~u_·~v =_k~u_{k · k}~v_{k ·}cos1200

= (2)(3)(₋1

2) = −3

❊①❡♠♣❧♦ ✸✳✾✳ ❙❡♥❞♦ k~u_k= 2 ✱ _k~v_k= 3 ❡1200

♦ â♥❣✉❧♦ ❡♥tr❡~u ❡~v✱ ❈❛❧❝✉❧❡ ~u_·~v✳

Pr♦♣r✐❡❞❛❞❡s✿

❉❛❞♦s ♦s ✈❡t♦r❡s~u ❡~v q✉❛s✐q✉❡r ❡ ♦s ♥ú♠❡r♦s r❡❛✐s α ❡ β✱ t❡♠♦s✿

❬E1 ❪❈♦♠✉t❛t✐✈✐❞❛❞❡✿ ~u_·~v =~v_·~u ❉❛ ❞❡✜♥✐çã♦ t❡♠♦s✿

~u_·~v=_k~u_{k · k}~v_{k ·}cos(~u, ~v) =_k~v_{k · k}~u_{k ·}cos(~v, ~u) =~v_·~u

❬E2 ❪Pr♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ r❡❧❛çã♦ à s♦♠❛✿

~u_·(~v+w~) = ~u_·~v+~u_·w~

❉❛ ✜❣✉r❛ 2.9 ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ✈❡t♦r~u ❝♦♠♦ s❡♥❞♦ ✉♥✐tár✐♦✱ t❡♠♦s✿

• kAC1k=k~v+w~k ·cos(~u, ~v+w~) =k~uk · k~v+w~k ·cos(~u, ~v+w~) =~u·(~v+w~)

• kAB1k=k~vk ·cos(~u, ~v) = k~uk · k~vk ·cos(~u, ~v) =~u·~v

❋✐❣✉r❛ ✸✳✶✵✿ Pr♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ E2

❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

• kB1C1k=kw~k ·cos(~u, ~w) =k~uk · kw~k ·cos(~u, ~w) =~u·w~

❈♦♠♦ kAC1k=kAB1k+kB1C1k✱ ❝♦♥❝❧✉í♠♦s q✉❡ ~u·(~v+w~) =~u·~v+~u·w~

❬E3 ❪ ❆ss♦❝✐❛t✐✈❛ ❡♠ r❡❧❛çã♦ ❛ ❡s❝❛❧❛r✿ α(~u~v) = (α~u)_·~v

α(~u~v) =α_k~u_{k · k}~v_{k ·}cos(~u, ~v) = _kα~u_{k · k}~v_{k ·}cos(α~u, ~v) = (α~u)_·~v❀

❬E4 ❪ ◆♦r♠❛✿ ~u_·~u=_k~u_k2

❈♦♠♦ ❛ ♠❡❞✐❞❛ ❡♥tr❡ ♦s â♥❣✉❧♦s ❞❡~u ❡ ~u é ③❡r♦✱ ❞❛ ❞❡✜♥✐çã♦✱ t❡♠♦s✿

~u_·~u=_k~u_{k · k}~u_{k ·}cos(~u, ~u) =_k~u_k2

·cos00

=_k~u_k2

❬E4 ❪ ❱❡t♦r❡s ❖r♦❣♦♥❛✐s✿ ~u_⊥~v _⇔~u_·~v = 0

❙❡ ✉♠ ❞♦s ✈❡t♦r❡s é ♥✉❧♦ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ é ✈❡r❞❛❞❡✱ s❡ ♥ã♦✱ t❡♠♦s✿

~u_·~v = 0_⇔cosθ = 0_⇔~u_⊥~v

❊①❡♠♣❧♦ ✸✳✶✵✳ Pr♦✈❛r✱ ✉t✐❧✐③❛♥❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ q✉❡ ♦ â♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠❛ s❡♠✐✲ ❝✐r❝✉♥❢❡rê♥❝✐❛ é r❡t♦✳

❈♦♥s✐❞❡r❛♥❞♦ ♦s ✈❡t♦r❡s~u ❡~v ❝♦♠♦ ♥❛ ✜❣✉r❛ ✸✳✶✶✱ ♦s ✈❡t♦r❡s~u+~v ❡~u₋~v ❞❡t❡r♠✐♥❛♠ ♦ â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡(~u+~v)_·(~u₋~v) = 0✱ t❡♠♦s✿

(~u+~v)_·(~u₋~v) = _k~u_k2

− k~v_k2

= 0

❋✐❣✉r❛ ✸✳✶✶✿ ➶♥❣✉❧♦ ✐♥s❝r✐t♦ ♥✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛ ❋♦♥t❡✿ Pró♣r✐♦ ❛✉t♦r

P♦✐s k~u_k=_k~v_k✭▼❡❞✐❞❛ ❞♦ r❛✐♦✮✳

Pr♦♣♦s✐çã♦✿ ❊♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ {u~1, ~v1, ~w1}✱ s❡ ~v = x1u~1 +y2v~1 +x3w~1 ❡ ~u =

x2u~1+y2v~1+z2w~1✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ ♦s ✈❡t♦r❡s ~u ❡~v é✿

~u_·~v =x1x2+y1y2+z1z2

✸✳✶✵ Pr♦❞✉t♦ ❱❡t♦r✐❛❧

❖ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s é ✉♠ ✈❡t♦r ❝✉❥❛ ♥♦r♠❛ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ✉♠❛ ♠❡❞✐❞❛ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✱ ♦✉ s❡❥❛✱ ✉♠❛ ár❡❛✳ ❚❡♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s à ❣❡♦♠❡tr✐❛ ❡♠ ♣r♦❜❧❡♠❛s ❝♦♠♦ ár❡❛s✱ ♣❡r♣❡♥❞✐❝✉❧❛r✐s♠♦s ❞❡ r❡t❛s ❡ ♣❧❛♥♦s✱ ❡♥tr❡ ♦✉tr♦s✳ ❖ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ t❛♠❜é♠ é ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ♥❛ ❞❡s❝r✐çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s ♥❛s ár❡❛s ❞❛ ▼❡❝â♥✐❝❛✱ ❊❧❡tr✐❝✐❞❛❞❡ ❡ ❊❧❡tr♦♠❛❣♥❡t✐s♠♦✳

❉❡✜♥✐çã♦ ✸✳✽ ✭Pr♦❞✉t♦ ❱❡t♦r✐❛❧✮✳ ❖ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s~u♣♦r~ué ♦ ✈❡t♦r ✐♥❞✐❝❛❞♦ ♣♦r~u_×~v ❞❡✜♥✐❞♦ ♣❡❧❛s s❡❣✉✐♥t❡s ❝❛r❛❝t❡ríst✐❝❛s✿

• ❉✐r❡çã♦✿ P❡r♣❡♥❞✐❝✉❧❛r ❛♦s ✈❡t♦r❡s ~ue~v✱ ♦✉ s❡❥❛~u_×~v _⊥~u ❡ ~u_×~v _⊥~v❀

• ◆♦r♠❛✿ k~u_×~v_k=_k~u_{k · k}~v_{k ·}sen(~u, ~v)❀

• ❙❡♥t✐❞♦✿ ➱ ❞❛❞♦ ♣❡❧❛ r❡❣r❛ ❞❛ ♠ã♦ ❞✐r❡✐t❛ q✉❡ é ❡q✉✐✈❛❧❡♥t❡✱ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❛ {~u, ~v, ~u_×~v_} s❡r ✉♠❛ ❜❛s❡ ♣♦s✐t✐✈❛ ❞♦R3✳

❊♠ r❡❧❛çã♦ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧✱ t❡♠♦s✿

• ❉✐r❡çã♦✿ ❙❡ tí✈❡ss❡♠♦s ❛♣❡♥❛s ❛ ❞✐r❡çã♦✱ t❡rí❛♠♦s ✐♥✜♥✐t❛s r❡♣r❡s❡♥t❛çõ❡s ♣❛r❛ ♦

✈❡t♦r~u_×~v✱ ♣♦✐s q✉❛❧q✉❡r ✈❡t♦r ♣❡rt❡❝❡♥t❡ à r❡t❛r✱ s❛t✐s❢❛③ ❛ ❞✐r❡çã♦ ❡①✐❣✐❞❛✭❋✐❣✉r❛

3.12✮✳