✷✵✶✻✿ ❉■❙❙❊❘❚❆➬➹❖ ❉❊ ▼❊❙❚❘❆❉❖
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❏♦ã♦ ❉❡❧✲❘❡✐ ✲ ❯❋❙❏ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❙❇▼
❘❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♦♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s ❡
❞❡t❡r♠✐♥❛♥t❡s
❆❧❡ss❛♥❞r❛ ❇❛r❜♦s❛ ❘♦❞r✐❣✉❡s ✶ ❋r❛♥❝✐♥✐❧❞♦ ◆♦❜r❡ ❋❡rr❡✐r❛✷
❘❡s✉♠♦✿ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❡♥s✐♥♦ ❞❡ ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐✲ ♥❛♥t❡s✱ ❛♣❧✐❝❛♥❞♦ s❡✉s ❝♦♥❝❡✐t♦s ❡ ♦♣❡r❛çõ❡s s♦❜ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦✳ P❛r❛ ✐st♦✱ ♣❛rt✐r❡♠♦s ❞❛ ♥♦çã♦ ❞❡ ✈❡t♦r✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♦♣❡r❛çõ❡s✱ ♣❛r❛ ❡♥tã♦✱ ♣♦r ♠❡✐♦ ❞❛ ❛ss♦❝✐❛çã♦ ❞❡ ❝❛❞❛ ❧✐♥❤❛ ❞❡ ✉♠❛ ♠❛tr✐③ à ✉♠ ✈❡t♦r✱ ♦❜t❡r♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♠❛tr✐③❡s ❡ s✉❛s ♦♣❡r❛çõ❡s✱ ❜❡♠ ❝♦♠♦ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❝♦♠♦ s❡♥❞♦ ❛ ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦✳ ◆♦ss♦ ❡st✉❞♦ s❡ r❡str✐♥❣✐rá ❛♦ ♣❧❛♥♦R2 ❡ ❛♦ ❡s♣❛ç♦ R3✱ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ❛♣❡♥❛s ❡st❡s sã♦ ❛♣❧✐❝❛❞♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❉❡t❡r♠✐♥❛♥t❡✱ ár❡❛ ❞❡ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ✈♦❧✉♠❡ ❞❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦✱ ár❡❛ ❞❡ tr✐â♥❣✉❧♦✳
✶ ■♥tr♦❞✉çã♦
❆♦ ❧♦♥❣♦ ❞❡ ♦♥③❡ ❛♥♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ t✉r♠❛s ❞❡ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ♦❜s❡r✈❛♥❞♦ ♦ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s ❡♠ r❡❧❛çã♦ ❛♦ ❡♥s✐♥♦ ❞❡ ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ♣✉❞❡ ♦❜s❡r✈❛r q✉❡ ♦ ❛❧✉♥♦ ♥ã♦ ❛ss✐♠✐❧❛✱ ❞❡ ♠♦❞♦ ❞❡s❡❥á✈❡❧✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s✱ tr✐✲ ❣♦♥♦♠❡tr✐❛ ❡ ❣❡♦♠❡tr✐❛✱ ❝♦♠ ♦s ❝á❧❝✉❧♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ♣♦✐s ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛❞♦t❛❞♦s tr❛③❡♠ ♦ ❝♦♥t❡ú❞♦ ❛♣❡♥❛s ❝♦♠♦ ✉♠❛ ✐♥tr♦❞✉çã♦ ♥❡❝❡ssár✐❛ à ❛♣❧✐❝❛çã♦ ❞❛ ❘❡❣r❛ ❞❡ ❈r❛♠❡r ♣❛r❛ s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ s❡♠ ♥❡♥❤✉♠❛ r❡❢❡rê♥❝✐❛ ❣❡♦♠étr✐❝❛ ❡ ❛♥❛❧ít✐❝❛ ❞❡ s✉❛s ♦♣❡r❛çõ❡s✳ ❯♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡st❡ ❢❛t♦ é ❛ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ♦ ❛❧✉♥♦ ❛♣r❡s❡♥t❛ ♣♦st❡r✐♦r♠❡♥t❡ ❡♠ ❝✉rs♦s ❞❡ ❣r❛❞✉❛çã♦ ❝♦♠ ❝♦♥t❡ú❞♦ ❞❡ á❧❣❡❜r❛ ❧✐♥❡❛r ❡ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✳ ❆♦ ❝♦♥trár✐♦ ❞♦ q✉❡ é ♠✐♥✐str❛❞♦ ❞❡♥tr♦ ❞❛s s❛❧❛s ❞❡ ❛✉❧❛ ❛❝❡r❝❛ ❞❡ss❡ ❛ss✉♥t♦✱ ❛ ♦r✐❡♥t❛çã♦ ❥á ❡①✐st❡♥t❡ ♣❡❧♦ ▼❊❈✱ ♥♦s P❈◆✬s✱ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s✱ ❞✐③ q✉❡✿
✶❆❧✉♥❛ ❞❡ ▼❡str❛❞♦ ❞♦ P❘❖❋▼❆❚✱ ❚✉r♠❛ ✷✵✶✹✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❏♦ã♦ ❉❡❧✲❘❡✐ ✲ ❯❋❙❏✱ ❛❧❡ss❛♥❞r❛❝♦q✷✵✶✵❅❤♦t♠❛✐❧✳❝♦♠
➱ ❞❡s❡❥á✈❡❧ t❛♠❜é♠ q✉❡ ♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ ❛❜♦r❞❡ ❝♦♠ s❡✉s ❛❧✉♥♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦ ✭❝♦❧❡çã♦ ❞♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❞❡ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✮ q✉❛♥t♦ ❛❧❣é❜r✐❝♦ ✭❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s✮✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ é ✐♠♣♦rt❛♥t❡ r❡❧❛❝✐♦♥❛r ❛s ♦♣❡r❛çõ❡s ❡①❡❝✉t❛❞❛s ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ✭s♦♠❛✱ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❡s❝❛❧❛r✮ ❝♦♠ s❡✉ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦✳ ❆ ✐♥❝❧✉sã♦ ❞❛ ♥♦çã♦ ❞❡ ✈❡t♦r ♥♦s t❡♠❛s ❛❜♦r❞❛❞♦s ♥❛s ❛✉❧❛s ❞❡ ♠❛t❡♠át✐❝❛ ✈✐r✐❛ ❝♦rr✐❣✐r ❛ ❞✐st♦rçã♦ ❝❛✉s❛❞❛ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ é ✉♠ tó♣✐❝♦ ♠❛t❡♠át✐❝♦ ✐♠♣♦rt❛♥t❡ ♠❛s q✉❡ ❡stá ♣r❡s❡♥t❡ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ s♦♠❡♥t❡ ♥❛s ❛✉❧❛s ❞❡ ❢ís✐❝❛✳ ✭✳✳✳✮ ❯♠❛ ✐♥tr♦❞✉çã♦ à ❣❡♦♠❡tr✐❛ ✈❡t♦r✐❛❧ ❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ ♣❧❛♥♦ ❡ ♥♦ ❡s♣❛ç♦✱ ✐s♦♠❡tr✐❛ ❡ ❤♦♠♦t❡t✐❛✱ é t❛♠❜é♠ ♠❛✐s ✉♠❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ tr❛❜❛❧❤❛r ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s s♦❜ ♦s ♣♦♥t♦s ❞❡ ✈✐st♦ ❛❧❣é❜r✐❝♦ ❡ ❣❡♦♠étr✐❝♦✳ ✭❙❡❣✉♥❞♦ ❬✶❪ ✱♣❛❣s✳ ✼✼ ❡ ✾✸✮
❚❡♥❞♦ ✐ss♦ ❡♠ ♠❡♥t❡✱ ♦ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❢❛③❡r ✉♠ ❜r❡✈❡ ❡st✉❞♦ s♦❜r❡ ❛❧❣✉♥s ❛ss✉♥t♦s q✉❡ ❞❡✈❡r✐❛♠ s❡r tr❛t❛❞♦s ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❞❡t❛❧❤❛❞❛ ❞❡♥tr♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ♣ú❜❧✐❝♦✳
◆❛ s❡çã♦ 2❢❛r❡♠♦s ✉♠❛ ❡①♣❧❛♥❛çã♦ s♦❜r❡ ✈❡t♦r❡s ✈♦❧t❛❞❛ à ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✳
◆❛ s❡çã♦ 3 tr❛t❛r❡♠♦s s♦❜r❡ ♠❛tr✐③❡s ❡ s✉❛s r❡♣r❡s❡♥t❛çõ❡s ❣❡♦♠étr✐❝❛s✱ ❛❜♦r❞❛r❡♠♦s t❛♠❜é♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ r♦t❛çã♦✱ ✉♠ t❡♠❛ ♥♦✈♦ ♣❛r❛ ♠✉✐t♦s ❛❧✉♥♦s✳
❋✐♥❛❧✐③❛♥❞♦ ♦ tr❛❜❛❧❤♦✱ ❛s ú❧t✐♠❛s s❡çõ❡s ❛❜♦r❞❛♠ s♦❜r❡ ❞❡t❡r♠✐♥❛♥t❡s ❞❡ s❡❣✉♥❞❛ ❡ t❡r❝❡✐r❛ ♦r❞❡♠ ❡ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞❡ ♠❛tr✐③❡s✳ ◆♦s ♣r✐♠❡✐r♦s✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ✈✐sã♦ ❣❡♦♠étr✐❝❛ ❞❡ s❡✉ s✐❣♥✐✜❝❛❞♦✱ ♦ q✉❡ t❛♠❜é♠ é r❛r❛♠❡♥t❡ ✈✐st❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❏á ♦ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞❡ ♠❛tr✐③❡s✱ é tr❛t❛❞♦ ❛q✉✐ ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❝á❧❝✉❧♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡✱ t❛♠❜é♠ ❛❜♦r❞❛❞♦ ❞❡ ❢♦r♠❛ ❣❡♦♠étr✐❝❛✳
❆ ♣r♦♣♦st❛ é tr❛❜❛❧❤❛r ❡st❡s ❝♦♥t❡ú❞♦s ♥♦ s❡❣✉♥❞♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♣♦✐s ♦ ❛❧✉♥♦ ❥á t❡rá ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ❢✉♥çõ❡s✱ tr✐❣♦♥♦♠❡tr✐❛ ❡ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✳
✷ ❆s ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s
❙❡❣✉♥❞♦ ❤✐st♦r✐❛❞♦r❡s✱ ❛ á❧❣❡❜r❛ ❧✐♥❡❛r é ✉♠ ❞♦s r❛♠♦s ♠❛✐s ❛♥t✐❣♦s ❞❛ ♠❛t❡♠át✐❝❛✳ ❋♦✐ ♥♦t❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥♦s ❡s❝r✐t♦s ❞♦s ♣♦✈♦s ❞❛ ❜❛❜✐❧ô♥✐❛ ❡♥tr❡ ✶✽✵✵ ❡ ✶✻✵✵ ❆❈✱ s❡❣✉♥❞♦ ❬✷❪✳ ❊♠❜♦r❛ ❡st❛ ár❡❛ t❡♥❤❛ s✉❛ ♦r✐❣❡♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♣rát✐❝❛✱ ❥á ❡r❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ r❡s♦❧✉çã♦ s✐♠✉❧tâ♥❡❛ ❞❡ ❡q✉❛çõ❡s ❡ ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s✳
❖s ❝❤✐♥❡s❡s✱ t❛♠❜é♠ ❞✉r❛♥t❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❥á ❤❛✈✐❛♠ ❡❧❛❜♦r❛❞♦ ✉♠ ♠ét♦❞♦ ♣❛r❛ r❡s♦✲ ❧✉çã♦ ❞❡ ✈ár✐❛s ❡q✉❛çõ❡s✱ ❝❤❛♠❛❞♦ ✏▼ét♦❞♦ ❋❛♥❣❝❤❡♥❣✑ q✉❡ é✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ♦ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ●❛✉ss✳
❆ á❧❣❡❜r❛ ❧✐♥❡❛r ♠❛✐s ♣❛r❡❝✐❞❛ ❝♦♠ ❛ q✉❡ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡✱ ❝♦♠❡ç♦✉ ❛ s❡r ❡st✉❞❛❞❛ ♣♦r ▲❡✐❜♥✐③ ✭✶✻✹✾ ✕ ✶✼✶✻✮✳ ❊❧❡ ✉s❛✈❛ ❝♦♠❜✐♥❛çõ❡s ❞❡ ❝♦❡✜❝✐❡♥t❡s ♣❛r❛ r❡s♦❧✈❡r ♦s s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ♦❜t❡♥❞♦ ✉♠❛ r❡❣r❛ ♣❛r❛ ♦ q✉❡ ❤♦❥❡ ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ ❞❡t❡r♠✐♥❛♥t❡✱ ♣♦ré♠ ❡ss❛ ♥♦♠❡♥❝❧❛t✉r❛ só ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ❡♠ ✶✽✵✶ ♣♦r ●❛✉ss✳ ❚❛♠❜é♠ ♥❡ss❛ é♣♦❝❛✱ ▼❛❝❧❛✉r✐♥ ✭✶✻✾✽ ✕ ✶✼✹✻✮✱ s❡❣✉♥❞♦ ❬✸❪✱ ❡s❝r❡✈❡✉ ♦ ❧✐✈r♦ ❯♠ tr❛t❛❞♦ s♦❜r❡ á❧❣❡❜r❛✱ ♦♥❞❡ é ❛❜♦r❞❛❞♦ s♦❜r❡ ♦ q✉❡ ❡❧❡ ♥♦♠❡♦✉ ❞❡ ✏♦ t❡♦r❡♠❛ ❣❡r❛❧✑✱ ❝♦♥❤❡❝✐❞♦ ❛t✉❛❧♠❡♥t❡ ❝♦♠♦ ❛ ❘❡❣r❛ ❞❡ ❈r❛♠❡r✳
❙❡❣✉♥❞♦ ❬✷❪✱ ❱❛♥❞❡r♠♦♥❞❡ ✭✶✼✸✺ ✕ ✶✼✾✻✮ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ❛♣r❡s❡♥t❛r ❞❡ ❢♦r♠❛ ❧ó❣✐❝❛ ❛ t❡♦r✐❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❡ tr❛tá✲❧❛ s❡♣❛r❛❞❛ ❞❛ s♦❧✉çã♦ ❞❡ s✐st❡♠❛ ❧✐♥❡❛r✳ ❆❧❣✉♥s ❛✉t♦r❡s ♦ ❝♦♥s✐❞❡r❛♠ ❝♦♠♦ ❢✉♥❞❛❞♦r ❞❛ t❡♦r✐❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s✳ ❊✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ▲❛♣❧❛❝❡ ✭✶✼✹✾ ✕ ✶✽✷✼✮ ♣r♦✈♦✉ ❛❧❣✉♠❛s ❞❡ s✉❛s r❡❣r❛s✳
▲❛❣r❛♥❣❡ ✭✶✼✸✻ ✕ ✶✽✶✸✮✱ ♣♦r ♠❡✐♦ ❞♦ ❡st✉❞♦ ❞♦s t❡tr❛❡❞r♦s ❝♦♥✈❡①♦s✱ ❡♥❝♦♥tr♦✉ ❛❧❣✉♠❛s ✐❞❡♥t✐❞❛❞❡s ❛❧❣é❜r✐❝❛s q✉❡ ♦ ❛❥✉❞❛r✐❛♠ ❛ ❡st❛❜❡❧❡❝❡r ✉♠ t❡♦r❡♠❛ s♦❜r❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✳ ●❛✉ss ✭✶✼✼✼ ✕ ✶✽✺✺✮✱ ❞❡♥tr❡ ✈ár✐❛s ❝♦♥tr✐❜✉✐çõ❡s✱ ❢♦✐ ♦ ♠❛✐♦r ❞✐✈✉❧❣❛❞♦r ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ♦✉✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❞❡ ♦✉tr❛s✱ ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳❱■■■✳ ❊❧❡ t❛♠❜é♠ ❢♦r♠✉❧♦✉ ♦ ✏▼ét♦❞♦ ❞❛ ❊❧✐♠✐♥❛çã♦✑✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ♦♣❡r❛çõ❡s às ❧✐♥❤❛s ❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❛ ✜♠ ❞❡ tr❛♥s❢♦r♠á✲❧❛ ❡♠ ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r s✉♣❡r✐♦r✳
❏á ❡♥tr❛♥❞♦ ♥❛ é♣♦❝❛ ❝♦♥t❡♠♣♦râ♥❡❛✱ ❈❛✉❝❤② ✭✶✼✽✾ ✕ ✶✽✺✼✮✱ s❡❣✉♥❞♦ ❬✸❪✱ ♣✉❜❧✐❝♦✉ ✉♠ ❞♦s ♠❡❧❤♦r❡s tr❛❜❛❧❤♦s s♦❜r❡ ♦ t❡♠❛✱ ♦♥❞❡ ❡❧❡ ❞❡♠♦♥str❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r♦❞✉t♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✳ ❊♠ ❬✷❪✱ ❙❝❤❡r❦ ✭✶✼✾✽ ✕ ✶✽✽✺✮ ♣r♦✈♦✉ q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r é ♦ ♣r♦❞✉t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ s✉❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✳ ❙②❧✈❡st❡r ✭✶✽✶✹ ✕ ✶✽✾✼✮ ❝♦♥❝❧✉✐✉ q✉❡✱ ♣❛r❛ q✉❡ ❞✉❛s ❡q✉❛çõ❡s✱ ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s✱ t✐✈❡ss❡♠ ❛ ♠❡s♠❛ s♦❧✉çã♦✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❢♦r♠❛❞❛ ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ss❛s ❡q✉❛çõ❡s✱ ❞❡✈❡r✐❛ s❡r ♥✉❧♦✳ ❈❛②❧❡② ✭✶✽✷✶ ✲✶✽✾✺✮ ❛♣r❡s❡♥t♦✉ ❛s ♠❛tr✐③❡s ❝♦♠♦ ❛s ❝♦♥❤❡❝❡♠♦s ❤♦❥❡✱ ❝♦♠ ❞✐♠❡♥sõ❡s m×n✱ ❡ ❞❡✜♥✐✉ s✉❛s ♦♣❡r❛çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s✳
❆t✉❛❧♠❡♥t❡✱ ❡①✐st❡♠ ♠✉✐t♦s ❡st✉❞♦s s♦❜r❡ ❛ t❡♦r✐❛ ❞❛s ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s ❡ ❝♦♠♦ ❞✐t♦ ❡♠ ❬✷❪✱ ✏é ✉♠ ❞♦s t❡♠❛s ♠❛✐s ❛♥t✐❣♦s ❡ ♠❛✐s ♥♦✈♦s ❞❛ ♠❛t❡♠át✐❝❛✑✳
✸ ▼❡t♦❞♦❧♦❣✐❛
◆❡st❛ s❡çã♦ ❛❜♦r❞❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r❡s✱ s✉❛s ♦♣❡r❛çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s✳
✸✳✶ ❱❡t♦r
❖s ✈❡t♦r❡s sã♦ s❡❣♠❡♥t♦s ❞❡ r❡t❛s ♦r✐❡♥t❛❞♦s✱ ♦✉ s❡❥❛✱ q✉❡ ♣♦ss✉❡♠ ✉♠ t❛♠❛♥❤♦✱ ❛q✉✐ ❝❤❛♠❛❞♦ ❞❡ ♠ó❞✉❧♦✱ ✉♠❛ ❞✐r❡çã♦ ❡ ✉♠ s❡♥t✐❞♦✳ P❛r❛ q✉❡ ✉♠ ✈❡t♦r s❡❥❛ ❞❡t❡r♠✐♥❛❞♦✱ é ♥❡❝❡ssár✐❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞♦✐s ♣♦♥t♦s✱ ♦ ✐♥í❝✐♦ ❡ tér♠✐♥♦ ❞♦ s❡❣♠❡♥t♦ ❞❡❧✐♠✐t❛❞♦ ♣♦r ❡❧❡s✳
◆♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ✉♠ ♣♦♥t♦ P é r❡♣r❡s❡♥t❛❞♦ ♣♦r s✉❛s ❝♦♦r❞❡♥❛❞❛s (x, y)✱ ❛s q✉❛✐s xé ♦ ✈❛❧♦r ❞❡ s✉❛ ❛❜s❝✐ss❛ ❡ y✱ ♦ ✈❛❧♦r ❞❡ s✉❛ ♦r❞❡♥❛❞❛✳ ❊st❡ ♣♦♥t♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ✈❡t♦r q✉❛♥❞♦ ❝♦♥s✐❞❡r❛r♠♦s ♦ ♣♦♥t♦ ❞❡ ✐♥í❝✐♦ ❝♦♠♦ s❡♥❞♦ ❛ ♦r✐❣❡♠ O = (0,0) ❞❡ss❡ ♣❧❛♥♦✳ ❖ ✈❡t♦r✱ ❛ss✐♠ ❝♦♠♦ ♦ ♣♦♥t♦✱ ♣♦ss✉✐ ❝♦♦r❞❡♥❛❞❛s✳ ❊❧❛s sã♦ ❞❡t❡r♠✐♥❛❞❛s ♣♦r✿
P −O = (x, y)−(0,0) = (x−0, y−0)
◆❡st❡ ❝❛s♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r OP~ s❡rá (x, y)✱ ❝♦✐♥❝✐❞✐♥❞♦ ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P✳
✸✳✶✳✶ ◆♦çõ❡s s♦❜r❡ ✈❡t♦r❡s
❊♠ s❡ tr❛t❛♥❞♦ ❞❡ ✈❡t♦r❡s ♥♦ ♣❧❛♥♦✱ s❡❣✉♥❞♦ ❬✹❪✱ ❡❧❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ~v = (a1, a2)✱ ♦♥❞❡a1 ❡ a2 sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ♦✉ ❝♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r✳
❯♠❛ ❜❛s❡ ✈❡t♦r✐❛❧ é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥st✐t✉í❞♦ ♣♦r ✈❡t♦r❡s ♣r✐♠ár✐♦s✱ ❞♦s q✉❛✐s ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♦✉tr♦s ✈❡t♦r❡s ♣♦r ♠❡✐♦ ❞❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❡♥tr❡ ❡❧❡s✱ ♦♥❞❡ ❡ss❛ ❝♦♠❜✐♥❛çã♦ é ✉♠❛ s♦♠❛ ❞♦s ✈❡t♦r❡s ♣r✐♠ár✐♦s ♠✉❧t✐♣❧✐❝❛❞♦s ♣♦r ❡s❝❛❧❛r❡s✳ ❋❛③❡♥❞♦✲s❡ ✉♠❛ ❛♥❛❧♦❣✐❛ à ❝♦♠❜✐♥❛çã♦ ❞❡ ❝♦r❡s✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❡ss❡s ✈❡t♦r❡s ♣r✐♠ár✐♦s às ❝♦r❡s ♣r✐♠ár✐❛s✱ ♦♥❞❡ ❞❛ ♠✐st✉r❛ ❡♥tr❡ ❡❧❛s sã♦ ♦❜t✐❞❛s ✈ár✐❛s ♦✉tr❛s ❝♦r❡s✳
❖ ♣♦♥t♦ P = (x, y) ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ s❡♥❞♦(x+ 0,0 +y)✱ ❝♦♠♦ ❞✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ q✉❡ ♣♦r s✉❛ ✈❡③ é ✐❣✉❛❧ ❛x(1,0)+y(0,1)✳ ❈❤❛♠❛♥❞♦✲s❡ ♦s ✈❡t♦r❡s(1,0) = e~1 ❡(0,1) = e~2✱
t❡r❡♠♦s✿
x(1,0) =x ~e1
y(0,1) =y ~e2
❉❛í✱ ♦ ✈❡t♦r (x, y) ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✱
(x, y) =x ~e1+y ~e2
❈♦♠♦ ❞✐t♦ ♥♦ ✐♥í❝✐♦ ❞❡st❛ s❡çã♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r e~1 ❡ e~2 ❝♦♠♦ ✈❡t♦r❡s ♣r✐♠ár✐♦s✱
♦s q✉❛✐s ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ R2✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❜❛s❡
❝❛♥ô♥✐❝❛✳ ✸
❊♠ r❡s✉♠♦✱ t♦♠❛♥❞♦ ✉♠ ✈❡t♦r(x, y)❞❡ss❡ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈ê✲❧♦ ❝♦♠♦ s❡♥❞♦x(1,0) +y(0,1)✳ P♦r ❡①❡♠♣❧♦✱ ♦ ✈❡t♦r (3,2)♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❝♦♠♦✱
(3,2) = 3(1,0) + 2(0,1)
➚ ❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❡s❝r❡✈❡r ✉♠ ✈❡t♦r ❝♦♠♦ (3,2) ❛ ♣❛rt✐r ❞❡ (1,0) ❡ (0,1)✱ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❡♥tr❡ ✈❡t♦r❡s✳ ❆ss✐♠ ✜❝❛ ❞❡t❡r♠✐♥❛❞♦ t♦❞♦ ✈❡t♦r (x, y) ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ♣♦✐s✱ ♦s ✐♥✜♥✐t♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦ ❞❡ss❡ ♣❧❛♥♦ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡e~1 ❡e~2✳
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ R2✱ q✉❛❧q✉❡r ♣♦♥t♦ P = (x, y, z)∈R3 ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠♦
✈❡t♦rOP~ ✱ s❡♥❞♦ O ♦ ♣♦♥t♦ (0,0,0)✳ P♦❞❡♠♦s t❛♠❜é♠ r❡❡s❝r❡✈❡r P = (x, y, z) ❝♦♠♦✿
P = (x, y, z) = x(1,0,0) +y(0,1,0) +z(0,0,1)
❊♥tã♦✱ ❝♦♠♦ ❡♠R2✱ e~1 = (1,0,0)✱e~2 = (0,1,0) ❡ e~3 = (0,0,1)✱ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛
R3✳
❆s ✜❣✉r❛s ✶✳❛✮ ❡ ✶✳❜✮ ♠♦str❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s r❡♣r❡s❡♥t❛çõ❡s ❣❡♦♠étr✐❝❛s ❞♦s ✈❡t♦r❡sOP~ = (x, y) ♥♦ ♣❧❛♥♦ ❡ OP~ = (x, y, z)♥♦ ❡s♣❛ç♦✳
✸❚❡r♠♦❧♦❣✐❛ ✉s❛❞❛ ♣❛r❛ ✐♥❞✐❝❛r ✉♠ ❡s♣❛ç♦ r❡❛❧ ❞❡ ❞✉❛s ❞✐♠❡♥sõ❡s✳
❋✐❣✉r❛ ✶✿ ❛✮OP~ ♥♦ ♣❧❛♥♦✱ ❜✮OP~ ♥♦ ❡s♣❛ç♦
❖ ♠ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r✱ q✉❡ ❡♠ ♠✉✐t❛s r❡❢❡rê♥❝✐❛s é tr❛t❛❞♦ t❛♠❜é♠ ❝♦♠♦ ♥♦r♠❛✱ é ♦ s❡✉ t❛♠❛♥❤♦✱ ♦✉ s❡❥❛✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ s❡✉s ♣♦♥t♦s ❡①tr❡♠♦s✳ ❆ ✜❣✉r❛ ✷ ❛ s❡❣✉✐r ♠♦str❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ✉♠ ✈❡t♦r~v❝♦♠ ♦r✐❣❡♠ ♥♦ ♣♦♥t♦A❡ ❡①tr❡♠✐❞❛❞❡ ♥♦ ♣♦♥t♦B✳
❋✐❣✉r❛ ✷✿ ▼ó❞✉❧♦ ❞♦ ✈❡t♦r~v
❚r❛ç❛♥❞♦ ♣❡❧♦ ♣♦♥t♦A✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦x❡ ♣♦r B ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦y✱ ♦❜t❡♠♦s ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ABC✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✷✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛
❞❡ P✐tá❣♦r❛s✹ ✱ t❡♠♦s✿
AB2 =AC2+BC2
❖✉ s❡❥❛✱
AB2 = (xB−xA)2+ (yB−yA)2
❊♥tã♦✱
AB =k~v k=p(xB−xA)2+ (yB−yA)2
❆s ❜❛s❡s ♠❛✐s ✉t✐❧✐③❛❞❛s sã♦ ❛s ♦rt♦♥♦r♠❛✐s✱ q✉❡ s❡ ❢♦r♠❛♠ q✉❛♥❞♦ s❡✉s ✈❡t♦r❡s ❢♦r❡♠ ♦rt♦❣♦♥❛✐s ❡ ✉♥✐tár✐♦s✱ ♦✉ s❡❥❛✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❡❧❡s s❡r ❞❡90◦
❡ s❡✉s ♠ó❞✉❧♦s t❡r❡♠ ✈❛❧♦r ✉♠✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❡♠♦s ❛ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦✿
❈♦♥s✐❞❡r❡ ✉♠❛ ❜❛s❡ ❞❡✜♥✐❞❛ ❝♦♠♦~e1, ~e2✳ ❊❧❛ é ❞✐t❛ ♦rt♦♥♦r♠❛❧ q✉❛♥❞♦~e1 ⊥~e2 ✭♦✉ s❡❥❛✱
~e1 é ♣❡r♣❡♥❞✐❝✉❧❛r ❛~e2✮ ❡ k~e1 k=k~e2 k= 1✳
✸✳✶✳✷ ❱❡t♦r❡s ❡q✉✐♣♦❧❡♥t❡s
❙❡❣✉♥❞♦ ❬✺❪✱ ❝❤❛♠❛♠♦s ❞❡ ✈❡t♦r❡s ❡q✉✐♣♦❧❡♥t❡s ❛ ❢❛♠í❧✐❛ ❞❡ t♦❞♦s ♦s ✈❡t♦r❡s q✉❡ ♣♦s✲ s✉❡♠ ♦s ♠❡s♠♦s ♠ó❞✉❧♦✱ s❡♥t✐❞♦ ❡ ❞✐r❡çã♦✳ ❆ ✜❣✉r❛ ❛ s❡❣✉✐r ♠♦str❛ ❛s r❡♣r❡s❡♥t❛çõ❡s ❣❡♦♠étr✐❝❛s ❞❡ três ✈❡t♦r❡s ❡q✉✐♣♦❧❡♥t❡s✳
✹◆❡ss❡ t❡♦r❡♠❛✱ P✐tá❣♦r❛s ❡♥✉♥❝✐♦✉ q✉❡ ♦ q✉❛❞r❛❞♦ ❞❛ ♠❡❞✐❞❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞❛s ♠❡❞✐❞❛s ❞❡ s❡✉s ❝❛t❡t♦s✳
❋✐❣✉r❛ ✸✿ ❋❛♠í❧✐❛ ❞❡ ✈❡t♦r❡s ❡q✉✐♣♦❧❡♥t❡s
❙❡♥❞♦✱
∡a =∡a′
=∡a′′ ✭❞✐r❡çã♦ ❞♦ ✈❡t♦r✮
k~v k=kv~′ k=kv~′′ k
❊ ♦ s❡♥t✐❞♦ ❞❛s ✢❡❝❤❛s✱ ♠♦str❛ q✉❡ t♦❞♦s ♦s ✈❡t♦r❡s ❡stã♦ ♥♦ ♠❡s♠♦ s❡♥t✐❞♦✳ ❆ s❡❣✉✐r ✈❡r❡♠♦s ❝♦♠♦ ❝♦♥str✉✐r ✉♠ ✈❡t♦rA~′
B′ ❡q✉✐♣♦❧❡♥t❡ ❛ ✉♠ ✈❡t♦r AB✿~
❚♦♠❡ ✉♠ ♣♦♥t♦ M ♥ã♦ ♣❡rt❡♥❝❡♥t❡ à r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦s A ❡ B✳ ▼❛rq✉❡ s♦❜r❡ ❛ r❡t❛ q✉❡ ❝♦♥tê♠ ♦s ♣♦♥t♦s A ❡ M✱ ✉♠ ♣♦♥t♦ B′✱ t❛❧ q✉❡
AM = M B′✳ ❉❡
♠♦❞♦ ❛♥á❧♦❣♦✱ ♠❛rq✉❡ ✉♠ ♣♦♥t♦A′ s♦❜r❡ ❛ r❡t❛ q✉❡ ❝♦♥tê♠ ♦s ♣♦♥t♦s
B ❡ M✱ t❛❧ q✉❡ BM =M A′✱ ✜❣✉r❛ ✹✳ ❖ ✈❡t♦r ~
A′
B′ é ❡q✉✐♣♦❧❡♥t❡ ❛♦ ✈❡t♦r AB✳~
❋✐❣✉r❛ ✹✿ ❙❡❣♠❡♥t♦s ❞❡ r❡t❛s ❡q✉✐♣♦❧❡♥t❡s
❉❡ ❢❛t♦✱ ♣♦r ❝♦♥str✉çã♦✱ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦s s❡❣♠❡♥t♦s A′
B ❡ AB′ ❡ ♦ ∡
AM B =
∡A′
M B′ sã♦ ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✳ ❚❡♠♦s ❡♥tã♦ ♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ▲❆▲ ✭▲❛❞♦ ➶♥❣✉❧♦
▲❛❞♦✮✱ ❧♦❣♦AB =A′
B′✱ ❛ss✐♠ ~
A′
B′ é ❡q✉✐♣♦❧❡♥t❡ àAB✳~
✸✳✶✳✸ ❖♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s
◆❡st❛ s❡çã♦ s❡rã♦ ❛❜♦r❞❛❞❛s ❛❧❣✉♠❛s ♦♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s✳
✶✳ ❙♦♠❛ ❞❡ ✈❡t♦r❡s
❉❛❞♦s ♦s ♣♦♥t♦s A✱ B ❡ C✱ ❢♦r♠❛♠✲s❡ ♦s ✈❡t♦r❡s ~v = AB~ ❡ w~ = BC✳ ❖ ✈❡t♦r~ ~v+w~ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ ✈❡t♦r ❝♦♠ ♦r✐❣❡♠ ♥♦ ♣♦♥t♦A❡ ❡①tr❡♠✐❞❛❞❡ ❡♠C✱ ✜❣✉r❛ ✺✳
❋✐❣✉r❛ ✺✿ ❙♦♠❛ ❞❡ ✈❡t♦r❡s
❉❛❞♦s~v ❡ w~ ✈❡t♦r❡s ❛r❜✐trár✐♦s✱ ♦ ✈❡t♦r~v+w~ é ❛ ❞✐❛❣♦♥❛❧ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ♦ q✉❛❧ é ♦❜t✐❞♦ ♣❡❧❛ s♦♠❛ ❞❡ ~v ❝♦♠ ✉♠ ✈❡t♦r ❡q✉✐♣♦❧❡♥t❡ à w~ ✱ ♦✉ ♣❡❧❛ s♦♠❛ ❞❡ w~ ❝♦♠ ✉♠ ✈❡t♦r ❡q✉✐♣♦❧❡♥t❡ à ~v ✱ ✜❣✉r❛ ✻✳
❋✐❣✉r❛ ✻✿ ❙♦♠❛ ❞❡ ✈❡t♦r❡s
P❛r❛ ♠❡❧❤♦r ✜①❛çã♦✱ ✐♠❛❣✐♥❡ q✉❡ ♥❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❡①✐st❛ ✉♠❛ t♦r♥❡✐r❛ q✉❡✱ q✉❛♥❞♦ ❛❜❡rt❛✱ ❛ á❣✉❛ ♣❡r❝♦rr❛ ♦ ✈❡t♦r~v ♣❛ss❛♥❞♦ ♣♦rw~ ❛té s✉❛ ❡①tr❡♠✐❞❛❞❡✳ P❛r❛ q✉❡ ❛ á❣✉❛ ✈á ❞❛ t♦r♥❡✐r❛ ✭✐♥í❝✐♦ ❞❡ ~v✮ ❛té ♦ ✜♥❛❧ ❞❡ w~ ♣❡r❝♦rr❡♥❞♦ ❛ ♠❡♥♦r ❞✐s✲ tâ♥❝✐❛✱ ❡❧❛ ♣r❡❝✐s❛r✐❛ ✐r ♣❡❧❛ t✉❜✉❧❛çã♦ ❞♦ ✈❡t♦r~v+w✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ s♦♠❛ ❞❡~ ~v❡w✳~
✷✳ ❉✐❢❡r❡♥ç❛ ❞❡ ✈❡t♦r❡s
P❛r❛ ❡❢❡t✉❛r♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ✈❡t♦r❡s~v ❡w~✱ ❜❛st❛ t♦♠❛r♠♦s ❛ s♦♠❛ ❞❡~v ❝♦♠ −w~ ✱ ♦✉ s❡❥❛✱ ♦ s❡✉ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ q✉❡✱ q✉❛♥❞♦ ❛♣❧✐❝❛❞♦ às ❣r❛♥❞❡③❛s ✈❡t♦r✐❛✐s✱ ✐♥✈❡rt❡ s❡✉ s❡♥t✐❞♦ ❡ s✉❛ ❞✐r❡çã♦✱ ❝♦♥s❡r✈❛♥❞♦ ♦ s❡✉ ♠ó❞✉❧♦✳ ❆ ✜❣✉r❛ ✼ ❛♣r❡s❡♥t❛ ❣r❛✜❝❛♠❡♥t❡ ♦ ✈❡t♦r~v−w~✳
❋✐❣✉r❛ ✼✿ ❉✐❢❡r❡♥ç❛ ❞❡ ✈❡t♦r❡s
❆ s♦♠❛ ❡ ❞✐❢❡r❡♥ç❛ ❞❡ ✈❡t♦r❡s s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭❛✮ ❈♦♠✉t❛t✐✈❛✿
~u+w~ =w~ +~u ∀~u, ~w∈R2
✭❜✮ ❆ss♦❝✐❛t✐✈❛✿
(~u+~v) +w~ =~u+ (~v+w~) ∀~u, ~v, ~w∈R2
✭❝✮ ❊❧❡♠❡♥t♦ ♥❡✉tr♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡ O= (0,0)∈R2✱ t❛❧ q✉❡✿
~u+ 0 = 0 +~u=~u ∀~u∈R2
✭❞✮ ■♥✈❡rs♦ ❛❞✐t✐✈♦✱ ❞❛❞♦ ~u∈R2✱ ❡①✐st❡ −~u∈R2✱ t❛❧ q✉❡✿
~u+ (−~u) =~u−~u= 0
❆s ♣r♦♣r✐❡❞❛❞❡s ❛♥t❡r✐♦r❡s ❝♦♥t✐♥✉❛♠ ✈á❧✐❞❛s s❡ s✉❜st✐t✉✐r♠♦s R2 ♣♦r R3✳
✸✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r❡s ♣♦r ❡s❝❛❧❛r❡s
❉❛❞♦ ✉♠ ✈❡t♦r ~v = (x, y) ∈ R2 ❡ ✉♠ ♥ú♠❡r♦ t ∈ R✱ t❡♠✲s❡ q✉❡ t~v = t(x, y) = (tx, ty)✳ ❊♠ ❬✺❪✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ✉♠ ❝❡rt♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ AA~ ′ q✉❛♥❞♦ ♠✉❧t✐✲
♣❧✐❝❛❞♦ ♣♦r ✉♠ ❡s❝❛❧❛r r❡❛❧✱ ❣❡r❛ ✉♠ ♦✉tr♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ AA~ ′′✱ t❛❧ q✉❡✿
kAA~ ′′
k=|t| kAA~ ′
k
❆ ✜❣✉r❛ ✽ ❛ s❡❣✉✐r ❛♣r❡s❡♥t❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦s ✈❡t♦r❡s~v✱ −2~v✱ 3~v ❡ −~v✳
❋✐❣✉r❛ ✽✿ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ❡s❝❛❧❛r r❡❛❧
❆ ♦♣❡r❛çã♦ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭❛✮ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿
s(t~v) = (st)~v ∀s, t∈R;~v ∈R2
✭❜✮ ❉✐str✐❜✉t✐✈✐❞❛❞❡✿
(s+t)~v =s~v+t~v ∀s, t∈R;~v ∈R2
t(~v+w~) =t~v+t ~w ∀t∈R;~v, ~w∈R2
❆s ♣r♦♣r✐❡❞❛❞❡s✱ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❞✐str✐❜✉t✐✈✐❞❛❞❡✱ ❛❝✐♠❛ ❝♦♥t✐♥✉❛♠ ✈á❧✐✲ ❞❛s s❡ s✉❜st✐t✉✐r♠♦sR2 ♣♦r R3✳
✹✳ Pr♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s
❙❡❣✉♥❞♦ ❬✻❪✱ ❞❛❞♦s ❞♦✐s ✈❡t♦r❡s ~v ❡ w~ ❞❡✜♥✐❞♦s ❡♠ R2 ♦✉ R3✱ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r
❡♥tr❡ ❡❧❡s é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♦r✐❣✐♥ár✐♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛s ♥♦r♠❛s ❞❡ss❡s ✈❡t♦r❡s ♣❡❧♦ ❝♦ss❡♥♦ ❞♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❡❧❡s✱ ✜❣✉r❛ ✾✱ ❞❡♥♦t❛❞♦ ❝♦♠♦✿
~v. ~w=k~vkkw~ kcosθ ✭✶✮
❋✐❣✉r❛ ✾✿ ➶♥❣✉❧♦ ❡♥tr❡ ✈❡t♦r❡s
❉❡ ✭✶✮ t❡♠♦s✱
cosθ = v.u
k~v k.k~uk
❈❛s♦ ✶✮ P❛r❛ ~v = 0 ♦✉ w~ = 0✱ t❡♠♦s q✉❡ ~v. ~w = 0✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ θ = 90◦✳
❈❛s♦ ✷✮ P❛r❛~v6= 0❡w~ 6= 0❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❈❛s♦ ✶✱ q✉❡~v.~v =k~v k2✱ ♦ â♥❣✉❧♦θ ❢♦r♠❛♥❞♦ ❡♥tr❡ ❡❧❡s ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ ▲❡✐ ❞♦s ❈♦ss❡♥♦s✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳
k(~v−w~)k2 =k~v k2+kw~ k2−2k~v k.kw~ kcosθ
❊♥tã♦✱
k~vk2−2~v ~w+kw~ k2 =k~v k2+kw~ k2−2k~v k.kw~ kcosθ
■s♦❧❛♥❞♦ ♦cosθ✱
cosθ= 2~v ~w− k~v k
2
− kw~ k2+k~vk2 +kw~ k2 2k~v k.kw~ k
❆ss✐♠✱
cosθ= kw~ +~vk
2
− k~vk2− kw~ k2 2k~v kkw~ k
❱♦❧t❛♥❞♦ ♥❛ ❡q✉❛çã♦(1) ❡ s✉❜st✐t✉✐♥❞♦ ♦cosθ ♣❡❧❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡✿
~v. ~w=k~v kkw~ k kw~ +~v k
2
− k~v k2− kw~ k2 2k~v kkw~ k
❊♥tã♦✱
~v. ~w= kw~ +~v k
2
− k~v k2− kw~ k2 2
❈♦♥s✐❞❡r❛♥❞♦ ♦s ✈❡t♦r❡s~v = (x1, y1, z1)❡ w~ = (x2, y2, z2)✱ ♦❜t❡r❡♠♦s✱
~v. ~w= (x1+x2) 2+ (y
1+y2)2+ (z1+z2)2−(x12+y12+z12)−(x22+y22+z22) 2
❖✉ ❛✐♥❞❛✱
~v. ~w= (x12+2x1x2+x22)+(y12+2y1y2+y22)+(z12+2z1z2+z22)−(x12+y12+z12)−(x22+y22+z22)
2
■st♦ é✱
~v. ~w= (x12+y12+z12)+(x22+y22+z22)+2(x1x2+y1y2+z1z2)−(x12+y12+z12)−(x22+y22+z22)
2
P♦rt❛♥t♦✱
~v. ~w= 2(x1x2+y1y2+z1z2)
2 =x1x2+y1y2+z1z2
◗✉❛♥❞♦ ~v= (x1, y1) ❡ w~ = (x2, y2)✱~v. ~w=x1x2+y1y2✳
❊♥tã♦✱ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ❞♦✐s ✈❡t♦r❡s é ❛ s♦♠❛ ❞❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ❡♥tr❡ s✉❛s ❝♦♦r❞❡♥❛❞❛s✳ ❱❡❥❛♠♦s ♦ ❡①❡♠♣❧♦✿
~v = (1,2) ❡w~ = (2,3)
❆ss✐♠✱ t❡♠♦s✱
~v. ~w= 8
❖ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❡ ❞♦✐s ✈❡t♦r❡s s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭❛✮ ❈♦♠✉t❛t✐✈✐❞❛❞❡✿
~v. ~w=w.~v~ ∀~v, ~w∈R2
(~u.~v). ~w=~v.(~u. ~w) = w.~ (~u.~v) ∀~u, ~v, ~w∈R2
✭❜✮ ❉✐str✐❜✉t✐✈✐❞❛❞❡✿
~u.(w~+~v) =~u. ~w+~u.~v ∀~u, ~v, ~w∈R2
❊st❛s ♣r♦♣r✐❡❞❛❞❡s ♣❡r♠❛♥❡❝❡♠ ✈á❧✐❞❛s ♣❛r❛~u✱~v ❡ w~ ∈R3✳
✸✳✷ ▼❛tr✐③❡s
❉❡✜♥❡✲s❡ ❝♦♠♦ ♠❛tr✐③✱ ✉♠❛ t❛❜❡❧❛ ❢♦r♠❛❞❛ ♣♦r ❡❧❡♠❡♥t♦s ❞✐str✐❜✉í❞♦s ❡♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ ❛ ♠❛tr✐③ é(m×n)✳
❈❛❞❛ ❡❧❡♠❡♥t♦ ❝♦♥t✐❞♦ ♥❡st❛ ♠❛tr✐③ é ♦r✐❣✐♥ár✐♦ ❞❡ ✉♠ í♥❞✐❝❡ ❞✉♣❧♦✱ s✐♠❜♦❧✐③❛❞♦ ♣♦r aij✱ ♦♥❞❡ir❡♣r❡s❡♥t❛ ❛ ❧✐♥❤❛ ❡j ❛ ❝♦❧✉♥❛ às q✉❛✐s ♣❡rt❡♥❝❡♠ ❡ss❡ ❡❧❡♠❡♥t♦✳ ❯♠❛ ♠❛tr✐③
M✱ ❝♦♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✱ é r❡♣r❡s❡♥t❛❞❛ ❡♥tã♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
M =
a11 a12 . . . a1n
a21 a22 . . . a2n
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
am1 am2 · · · amn
❆ s❡❣✉✐r ✈❛♠♦s ❛ss♦❝✐❛r ❛ ✉♠❛ ♠❛tr✐③ M s❡✉s ✈❡t♦r❡s✲❧✐♥❤❛✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ♠❛tr✐③✿
M =
1 3 4 2
❈♦♥s✐❞❡r❛♥❞♦ ❝❛❞❛ ❧✐♥❤❛ ❞❡ M ❝♦♠♦ ✈❡t♦r✱ t❡♠♦s ~v = (1,3) ❡ w~ = (4,2)✳ ▲❡♠❜r❛♥❞♦ q✉❡ ❛♠❜♦s tê♠ ♦r✐❣❡♠ ❡♠O = (0,0)✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡ss❛ ♠❛tr✐③ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♣❡❧♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈❡t♦r❡s~v ❡ w~✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✶✵ ✿
❋✐❣✉r❛ ✶✵✿ ❘❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♠❛tr✐③
❖s ❡❧❡♠❡♥t♦s ❞❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❡✐①♦ ❞❛s ❛❜❝✐ss❛s ❞♦s ✈❡t♦r❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ s❡❣✉♥❞❛ ❝♦❧✉♥❛✱ ❞♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✳
❖❜s❡r✈❡ q✉❡✱ ❛♦ r❡♣r❡s❡♥t❛r ♦s ✈❡t♦r❡s ❢♦r♠❛❞♦s ♣❡❧❛s ❧✐♥❤❛s ❞❡ss❛ ♠❛tr✐③ ♥♦ ♣❧❛♥♦✱ t❡♠♦s ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ q✉❛♥❞♦ ♦s ✈❡t♦r❡s ♥ã♦ ❢♦r❡♠ ♠ú❧t✐♣❧♦s ✉♠ ❞♦ ♦✉tr♦✳ ◆❡st❡ ❝❛s♦✱ ❡ss❛ ♠❛tr✐③ é ❝❤❛♠❛❞❛ q✉❛❞r❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ❛ ♠❛tr✐③ q✉❛❞r❛ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ é ❛q✉❡❧❛ q✉❡ ♣♦ss✉✐ três ✈❡t♦r❡s✲❧✐♥❤❛ ❡♠R3✱ ❡st❡s ❢♦r♠❛rã♦ ✉♠
♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❛♥❞♦ ♥❡♥❤✉♠ ❞❡st❡s ✈❡t♦r❡s✲❧✐♥❤❛ ❢♦r❡♠ ❞❡♣❡♥❞❡♥t❡s ❞♦s ❞❡♠❛✐s✳ ◆♦ss♦ ❡st✉❞♦ s❡ r❡str✐♥❣✐rá à ❡st❡s ❞♦✐s t✐♣♦s ❞❡ ♠❛tr✐③❡s q✉❛♥❞♦✱ ♣♦st❡r✐♦r♠❡♥t❡✱ tr❛t❛r♠♦s ❞♦ ❝á❧❝✉❧♦ ❡ ❞❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s✳
✸✳✷✳✶ ❚✐♣♦s ❞❡ ♠❛tr✐③❡s
◆❡st❛ s❡çã♦✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❡①♣❧❛♥❛çã♦ s♦❜r❡ ♦s ♣r✐♥❝✐♣❛✐s t✐♣♦s ❞❡ ♠❛tr✐③❡s ♣❛r❛ ✐♥✐❝✐❛r♠♦s ♣♦st❡r✐♦r♠❡♥t❡ s✉❛s ♦♣❡r❛çõ❡s✳
✶✳ ▼❛tr✐③ ❧✐♥❤❛
➱ ❛ ♠❛tr✐③ ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣♦r ✉♠❛ ❧✐♥❤❛✱ ♦✉ s❡❥❛✱ ❞❡✜♥✐❞❛ ❛♣❡♥❛s ❝♦♠ t❡r♠♦s a1j✱ ❝♦♠ 1 ≤ j ≤ n✳ ❊❧❛ ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r s❡r ✉♠ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ ❛♦
♣❧❛♥♦✱ q✉❛♥❞♦ n= 2✱ ♦✉ ❛♦ ❡s♣❛ç♦✱ q✉❛♥❞♦ n= 3✳
❈♦♠♦ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ♠❛tr✐③✿
M = 2 5 3
◆❡st❡ ❝❛s♦ t❡♠♦s ♦ ✈❡t♦r✲❧✐♥❤❛~v = (2,5,3)✳ ◆❛ ✜❣✉r❛ ✶✶ t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛✲ çã♦ ❣❡♦♠étr✐❝❛ ❞❡ss❛ ♠❛tr✐③✳
❋✐❣✉r❛ ✶✶✿ ▼❛tr✐③ ❧✐♥❤❛
✷✳ ▼❛tr✐③ ❝♦❧✉♥❛
➱ ❛ ♠❛tr✐③ ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣♦r ✉♠❛ ❝♦❧✉♥❛✱ ♦✉ s❡❥❛✱ só ♣♦r t❡r♠♦s ai1✱ ❝♦♠
1≤i≤m✳
❊ss❛ ♠❛tr✐③ s❡rá r❡♣r❡s❡♥t❛❞❛ ❡♠ ✉♠❛ r❡t❛✱ ♣♦✐s s❡✉s ✈❡t♦r❡s✲❧✐♥❤❛ ♣♦ss✉❡♠ ❛♣❡✲ ♥❛s ✉♠❛ ❞✐♠❡♥sã♦✱ ♦✉ s❡❥❛✱ ♣❡rt❡♥❝❡♥t❡s à R✳ ◆♦t❛✲s❡ t❛♠❜é♠ q✉❡✱ ♣♦r ♣♦ss✉ír❡♠
❡ss❛ ❝❛r❛❝t❡ríst✐❝❛✱ ❡ss❡s ✈❡t♦r❡s sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✱ ♦✉ s❡❥❛✱ sã♦ ♠ú❧t✐♣❧♦s✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ♠❛tr✐③✱
M =
1 2 3
◆❡st❡ ❝❛s♦✱ ❢♦r♠❛♠♦s ♦s ✈❡t♦r❡s ~u = (1)✱ ~v = (2) ❡ w~ = (3)✳ ◆❛ ✜❣✉r❛ ❛ s❡❣✉✐r t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ss❛ ♠❛tr✐③✳
❋✐❣✉r❛ ✶✷✿ ▼❛tr✐③ ❝♦❧✉♥❛
✸✳ ▼❛tr✐③ q✉❛❞r❛❞❛
P❛r❛ q✉❡ ✉♠❛ ♠❛tr✐③ s❡❥❛ q✉❛❞r❛❞❛✱ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❡✈❡ s❡r ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s✳
❱❡❥❛ ❛ s❡❣✉✐r ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠2✳
M =
1 0 0 1
❋✐❣✉r❛ ✶✸✿ ▼❛tr✐③ q✉❛❞r❛❞❛
◆❡st❡ ❝❛s♦✱ ♦s ✈❡t♦r❡s✲❧✐♥❤❛ sã♦~v = (1,0)❡ w~ = (0,1)✳
◆♦t❡ q✉❡✱ ❛♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ s♦♠❛ ❞❡ ✈❡t♦r❡s ❡♥✉♥❝✐❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ~v+w~ = (1,1)✱ ❡ t❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦ ❡ t❛♠❛♥❤♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ss❡ q✉❛❞r✐❧át❡r♦✳
❆❣♦r❛ ✈❡❥❛♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ 3✳
M =
1 0 0 0 1 0 0 0 1
◆❡st❡ ❝❛s♦✱ ~u = (1,0,0)✱ ~v = (0,1,0) ❡ w~ = (0,0,1) sã♦ s❡✉s ✈❡t♦r❡s✲❧✐♥❤❛ ❡✱ ❞❡ ♠❛♥❡✐r❛ s❡♠❡❧❤❛♥t❡ à ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ~u+~v +w~ = (1,1,1)✱ t❡♠ ♠❡s♠❛ ❞✐r❡çã♦ ❡ t❛♠❛♥❤♦ ❞❛ ❞✐❛❣♦♥❛❧ OP~ ✱ ❛❣♦r❛ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦✱ ✜❣✉r❛ ✶✹✳ ◆♦t❡ q✉❡ ~u=e~1✱~v =e~2 ❡ w~ =e~3✳
❋✐❣✉r❛ ✶✹✿ ▼❛tr✐③ ✐❞❡♥t✐❞❛❞❡
◆❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱ ♦❜s❡r✈❡ q✉❡ s❡✉s ✈❡t♦r❡s✲❧✐♥❤❛ sã♦ ♦rt♦♥♦r♠❛✐s✱ ♦✉ s❡❥❛✱ sã♦ ♦rt♦❣♦♥❛✐s ❡ s✉❛ ♥♦r♠❛ é 1✳
✹✳ ▼❛tr✐③ ❞✐❛❣♦♥❛❧
❈♦♠ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱ ♣♦❞❡♠♦s ♦❜t❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠é✲ tr✐❝❛ ❞❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❞❡✜♥✐❞❛ ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦✿
▼❛tr✐③ ❞✐❛❣♦♥❛❧ s✉r❣❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❝❛❞❛ ✈❡t♦r✲❧✐♥❤❛ ❞❡ ✉♠❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ♣♦r ❝❛❞❛ ❡s❝❛❧❛rkp,❝♦♠1≤p≤n✱ ♦♥❞❡n é ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③
q✉❛❞r❛❞❛✳ ❖✉ s❡❥❛✿
k1 0 . . . 0
0 k2 . . . 0
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
0 0 · · · kn
◆♦ ❝❛s♦ ❞❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠✱ t❡♠♦s✱
M =
k1 0 0
0 k2 0
0 0 k3
◆❡st❡ ❝❛s♦ ❛s ❧✐♥❤❛s ❞❡ M sã♦ ✐❞❡♥t✐✜❝❛❞❛s ♣♦r✿
~u= (k1,0,0) ♦✉ ~u=k1(1,0,0)
~v= (0, k2,0) ♦✉ ~v=k2(0,1,0)
~
w= (0,0, k3) ♦✉ w~ =k3(0,0,1)
❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡st❛ ♠❛tr✐③ s❡ ❡♥❝♦♥tr❛ ♥❛ ✜❣✉r❛ ✶✺✳
❋✐❣✉r❛ ✶✺✿ ▼❛tr✐③ ❞✐❛❣♦♥❛❧
❈♦♠♦ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ♠❛tr✐③✱
M =
2 0 0 0 3 0 0 0 4
❖♥❞❡ k1 = 2✱ k2 = 3 ❡ k3 = 4✳ ❈♦♥s✐❞❡r❛♥❞♦ ~u = (1,0,0)✱ ~v = (0,1,0) ❡
~
w= (0,0,1)✱ ♦ ✈❡t♦r OP~ ✱ ❞✐❛❣♦♥❛❧ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡t❡r♠✐♥❛❞♦ ~u✱~v ❡w✱ é ❞❛❞♦~ ♣♦r OP~ = 2~u+ 3~v+ 4w✳ ❆ s❡❣✉✐r t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ♠❛tr✐③~ M✳
❋✐❣✉r❛ ✶✻✿ ❊①❡♠♣❧♦ ❞❡ ♠❛tr✐③ ❞✐❛❣♦♥❛❧
✺✳ ▼❛tr✐③ ♥ã♦✲q✉❛❞r❛❞❛ ❉❡✜♥✐çã♦✿
❯♠❛ ♠❛tr✐③m×n é ❞✐t❛ ♥ã♦✲q✉❛❞r❛❞❛ q✉❛♥❞♦ s❡✉ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s é ❞✐❢❡r❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ s✉❛s ❝♦❧✉♥❛s✱ ♦✉ s❡❥❛✱
n6=m
❈♦♠♦ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ♠❛tr✐③ 3×2 ❞❛❞❛ ♣♦r✿
M =
3 1 1 1 1 4
❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ss❛ ♠❛tr✐③ ❡♥❝♦♥tr❛✲s❡ ♥❛ ✜❣✉r❛ ✶✼✱ ❛ s❡❣✉✐r✱
❋✐❣✉r❛ ✶✼✿ ▼❛tr✐③ ♥ã♦✲q✉❛❞r❛❞❛
✸✳✷✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ✈ár✐❛s ♦♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s ❡ s✉❛s r❡♣r❡s❡♥t❛çõ❡s ❣❡♦✲ ♠étr✐❝❛s✳
✶✳ ❆❞✐çã♦ ❡ s✉❜tr❛çã♦ ❞❡ ♠❛tr✐③❡s
❈♦♥s✐❞❡r❡♠♦s ♦s ✈❡t♦r❡s~v ❡w~ ❝♦♠ ♠❡s♠❛ ♦r✐❣❡♠✳ ❆♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ♣♦r ❡s❝❛❧❛✲ r❡s ❡ s♦♠❛r♠♦s✱ tr❛♥s❢♦r♠❛♠♦✲♦s ❡♠ ♥♦✈♦s ✈❡t♦r❡s ❝♦♠ ❞✐r❡çõ❡s✱ s❡♥t✐❞♦s ❡ ♠ó❞✉❧♦s ❞✐❢❡r❡♥t❡s ❞♦s ✐♥✐❝✐❛✐s✳ ❯s❛r❡♠♦s ❡ss❛s ✐❞❡✐❛s ♣❛r❛ tr❛t❛r♠♦s ❞❛ s♦♠❛ ❡ s✉❜tr❛çã♦ ❞❡ ♠❛tr✐③❡s✳
P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ s♦♠❛r♠♦s ❛s ♠❛tr✐③❡sA ❡ B✱ ❞❛❞❛s ♣♦r✿
A=
1 0 0 1
B =
1 4 3 1
❚♦♠❛♠♦s ♦s ✈❡t♦r❡s ~v = (1,0) ❡ w~ = (0,1)✱ q✉❡ sã♦ ❛s ❧✐♥❤❛s ❞❛ ♠❛tr✐③ A✱ ❡ ♦s ✈❡t♦r❡s v~′ = (1,4) ❡ w~′ = (3,1)✱ ❛s ❧✐♥❤❛s ❞❛ ♠❛tr✐③ B✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛s s♦♠❛s
~v+v~′ = (2,4)❡ w~ +w~′ = (3,2)✱ ♦❜té♠✲s❡ ❛ ♠❛tr✐③✿
A+B =
2 4 3 2
❆ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ♠❛tr✐③ A+B✱ ❡♥❝♦♥tr❛✲s❡ ♥❛ ✜❣✉r❛ ✶✽✳
❋✐❣✉r❛ ✶✽✿ ❙♦♠❛ ❞❡ ♠❛tr✐③❡s
❖♥❞❡ ♦s ✈❡t♦r❡s ✈❡r♠❡❧❤♦s sã♦ ♦r✐✉♥❞♦s ❞❛ ♣r✐♠❡✐r❛ ♠❛tr✐③ ❞❛ s♦♠❛ ❡ ♦s ❛③✉✐s✱ ♣r♦✈ê♠ ❞❛ s❡❣✉♥❞❛ ♠❛tr✐③✳ ❖s ♣r❡t♦s r❡♣r❡s❡♥t❛♠ ❛ ♠❛tr✐③ s♦♠❛✳ ◆♦t❡ q✉❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ú❧t✐♠♦s ✈❡t♦r❡s ❝✐t❛❞♦s✱ s✉r❣✐r❛♠ ❞❛ s♦♠❛ ❞❡ ♦✉tr♦s ❞♦✐s✱ ✉♠ ❛③✉❧ ❡ ♦✉tr♦ ✈❡r♠❡❧❤♦✳
❉❡ ♠❛♥❡✐r❛ ♠❛✐s ❢♦r♠❛❧✱ ❝♦♠♦ ❡♠ ❬✼❪✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛❞✐çã♦ ❡ s✉❜tr❛çã♦ ❞❡ ♠❛tr✐③❡s ❝♦♠♦ ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦✿
❆ s♦♠❛ ♦✉ ❞✐❢❡r❡♥ç❛ ❞❛s ♠❛tr✐③❡s A = (aij)m×n ❡ B = (bij)m×n✳ ❖✉ s❡❥❛✱ ❛♠❜❛s
❝♦♠ ❛s ♠❡s♠❛s ❞✐♠❡♥sõ❡s✱ ♦r✐❣✐♥❛ ✉♠❛ ♠❛tr✐③ C = (cij)m×n✱ ❝✉❥♦s t❡r♠♦s sã♦
♦❜t✐❞♦s s♦♠❛♥❞♦✲s❡ ♦✉ s✉❜tr❛✐♥❞♦✲s❡ ♦s ❡❧❡♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❡ A ❡ B✿
cij =aij ±bij ♣❛r❛ 1≤i≤m e 1≤j ≤n
✷✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③ ♣♦r ❡s❝❛❧❛r
▼✉❧t✐♣❧✐❝❛r ✉♠❛ ♠❛tr✐③ ♣♦r ✉♠ ❡s❝❛❧❛r k ∈ R✱ é ♠✉❧t✐♣❧✐❝❛r ♣♦r k s❡✉s ✈❡t♦r❡s✲ ❧✐♥❤❛✱ ♦✉ s❡❥❛✱
k.
a11 a12
a21 a22
=
ka11 ka12
ka21 ka22
❱❛❧❡ ❞❡st❛❝❛r q✉❡✱ ♠❡s♠♦ ❛♣ós ❡ss❛ tr❛♥s❢♦r♠❛çã♦✱ s❡✉s ✈❡t♦r❡s✲❧✐♥❤❛ ♠❛♥t❡♠ ♦s ♠❡s♠♦s s❡♥t✐❞♦s ❡ ❞✐r❡çõ❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ k > 0✳ ❈♦♠ r❡❧❛çã♦ ❛♦ ♠ó❞✉❧♦✱ ✈❛❧❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❛ s❡❣✉✐r✿
❙❡❥❛ ~u ✉♠ ❞♦s ✈❡t♦r❡s✲❧✐♥❤❛ ❞❛ ♠❛tr✐③ ✐♥✐❝✐❛❧ ❡ ~v ♦ ✈❡t♦r✲❧✐♥❤❛ ♦r✐❣✐♥ár✐♦ ❞❡ ~u ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧♦ ❡s❝❛❧❛r k✱ ♦✉ s❡❥❛✱~v =k~u✳ ❊♥tã♦✿
k~v k<k~uk quando −1< k <1
k~v k>k~uk quando k <−1 ou k > 1
k~v k=k~uk quando k= 1
P❛r❛ ❡①❡♠♣❧✐✜❝❛r ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ♠❛tr✐③ A ♣♦r ✉♠ ❡s❝❛❧❛r k✱ ❝♦♥s✐❞❡r❡✱
A=
1 2 3 1
❡ k = 2
■♥✐❝✐❛❧♠❡♥t❡ t❡♠♦s ♦s ✈❡t♦r❡s✲❧✐♥❤❛~v = (1,2) ❡ w~ = (3,1)✳ ❆♦ ♠✉❧t✐♣❧✐❝á✲❧♦s ♣♦r 2✱ ♦❜t❡♠♦s ♦s s❡❣✉✐♥t❡s ✈❡t♦r❡s✲❧✐♥❤❛ v~′ = 2(1,2) = (2,4) ❡ w~′ = 2(3,1) = (6,2)✱
❢♦r♠❛♥❞♦ ❛ ♠❛tr✐③ A′✳
A′
= 2A=
2 4 6 2
◆❛ ✜❣✉r❛ ✶✾✳❛✮ ❛ s❡❣✉✐r✱ ❡stã♦ r❡♣r❡s❡♥t❛❞❛s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❛s ♠❛tr✐③❡s A ❡ A′✳ ◆❛ ✜❣✉r❛ ✶✾✳❜✮✱ ❡♥❝♦♥tr❛♠✲s❡ ❛s r❡♣r❡s❡♥t❛çõ❡s ❣❡♦♠étr✐❝❛s ❞❛s ♠❛tr✐③❡s
A ❡ (−1)A✳
(−1)A=
−1 −2
−3 −1
❋✐❣✉r❛ ✶✾✿ ❛✮ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③ ♣♦rk = 2❀ ❜✮ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③ ♣♦r k=−1
✸✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❡♥tr❡ ❞✉❛s ♠❛tr✐③❡s
❆ s❡❣✉✐r ✐♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✱ s❡❣✉♥❞♦ ❬✽❪✳
❉❡✜♥✐çã♦✿
❈♦♥s✐❞❡r❡ ❞✉❛s ♠❛tr✐③❡s✱ A ❡ B✱ t❛✐s q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❡ A é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❡ B✱ ♦✉ s❡❥❛ A= (aij)m×p ❡B = (bij)p×n✱ ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❡❧❛s✱
A.B✱ é ❡①♣r❡ss♦ ♣♦r✿
cij =ai1b1j +ai2b2j+. . .+aipbpj ♣❛r❛ 1≤i≤m e 1≤j ≤n
❖✉ s❡❥❛✱ [AB]ij =ai1b1j +ai2b2j+. . .+aipbpj
❆ ♠❛tr✐③C = (cij)m×n✱ ♦r✐❣✐♥ár✐❛ ❞❡ss❡ ♣r♦❞✉t♦✱ ♣♦ss✉✐ ❞✐♠❡♥sõ❡s m × n✳
❖✉ s❡❥❛✱ ♣❛r❛ s❡ ♦❜t❡r ♦ ❡❧❡♠❡♥t♦ cij✱ ♠✉❧t✐♣❧✐❝❛✲s❡ ♦r❞❡♥❛❞❛♠❡♥t❡ ♦ ♣r✐♠❡✐r♦
❡❧❡♠❡♥t♦ ❞❛ ❧✐♥❤❛ ❞❛ ♠❛tr✐③ A ♣❡❧♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ B✱ ♦ s❡❣✉♥❞♦ ❡❧❡♠❡♥t♦ ❞❡ss❛ ♠❡s♠❛ ❧✐♥❤❛ ❞❡A❝♦♠ ♦ s❡❣✉♥❞♦ ❡❧❡♠❡♥t♦ ❞❛ ♠❡s♠❛ ❝♦❧✉♥❛ ❞❡ B✱ r❡♣❡t✐♥❞♦✲s❡ ❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦ ❛té ♦ ú❧t✐♠♦ ❡❧❡♠❡♥t♦ ❞❛ ❧✐♥❤❛ ❞❡ A ❡ ❝♦❧✉♥❛ ❞❡ B✱ s♦♠❛♥❞♦✲s❡ t♦❞❛s ❡ss❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ❛♦ ✜♥❛❧✳
❉❡ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✱ ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❞✉❛s ♠❛tr✐③❡s ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳
a11 a12 . . . a1p
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ai1 ai2 . . . aip
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
am1 am2 · · · amp
.
b11 . . . b1j . . . b1n
b21 . . . b2j . . . b2n
✳✳✳ . . . ✳✳✳ . . . ✳✳✳
bp1 · · · bpj · · · bpm
=
c11 . . . c1n
✳✳✳ cij ✳✳✳
cm1 · · · bmn
❉❛❞❛s ❛s ♠❛tr✐③❡s M✱ N ❡ P✱ ❡ ❛❞♠✐t✐♥❞♦ ❛s s♦♠❛s ❡ ♠✉❧t✐♣❧✐❝❛çõ❡s✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭❛✮ ❆ss♦❝✐❛t✐✈❛✿
(M N)P =M(N P) ✭❜✮ ❉✐str✐❜✉t✐✈❛✿
(M +N)P =M P +N P M(N +P) =M N +M P
❱❛❧❡ ❞❡st❛❝❛r✱ s❡❣✉♥❞♦ ❬✺❪✱ q✉❡ ❡①✐st❡♠ q✉❛tr♦ ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ♦ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s ❡ ♦ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r✳
✶❛✮ ❖ ♣r♦❞✉t♦ ❡♥tr❡ ♠❛tr✐③❡s só é ❞❡✜♥✐❞♦ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ ♣r✐♠❡✐r❛
é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ s❡❣✉♥❞❛✳
✷❛✮ ◆ã♦ ✈❛❧❡✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♥❡♠ s❡♠✲
♣r❡ M N s❡rá ✐❣✉❛❧ ❛N M✳ ❱❡❥❛ ❛ s❡❣✉✐r ✉♠ ❡①❡♠♣❧♦ ❞❡ ❞✉❛s ♠❛tr✐③❡s✱A ❡B✱ t❛✐s q✉❡ AB 6=BA✳
0 1 2 0
| {z }
A
0 1 1 0
| {z }
B = 1 0 0 2 0 1 1 0
| {z }
B
0 1 2 0
| {z }
✸❛✮ ❯♠ ❢❛t♦ ✐♠♣♦rt❛♥t❡ ❡♠ r❡❧❛çã♦ à ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s é q✉❡ ♣♦❞❡♠♦s
t❡r ❞✉❛s ♠❛tr✐③❡s ♥ã♦✲♥✉❧❛s ❝♦♠ ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❡❧❛s s❡♥❞♦ ✉♠❛ ♠❛tr✐③ ♥✉❧❛✱ ♣♦❞❡♠♦s ✐♥❝❧✉s✐✈❡ t❡r M2 = 0✱ ❝♦♠ M 6= 0✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦✳
2 2
−2 −2
2 2
−2 −2
= 0 0 0 0
❆s ♣r♦♣r✐❡❞❛❞❡s ❛♥t❡r✐♦r❡s ♥ã♦ ✈❛❧❡♠ ♣❛r❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♥tr❡ ♥ú♠❡r♦s r❡❛✐s✳ ✹❛✮ ◗✉❛♥❞♦ AB =BA =I
n✱ ❞✐③❡♠♦s q✉❡ ✉♠❛ ♠❛tr✐③ é ❛ ✐♥✈❡rs❛ ❞❛ ♦✉tr❛✳
❚♦❞♦ ♥ú♠❡r♦ r❡❛❧ k✱ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ♣♦ss✉✐ s❡✉ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✿ k−1
= 1
k✱ ♣♦✐s k(k
−1
) = (k−1
)k = 1✳ ▼❛s✱ ♣❛r❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ Mn×n✱
♥❡♠ s❡♠♣r❡ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ Pn×n✱ t❛❧ q✉❡ M P = P M = In✳ ◗✉❛♥❞♦ ❛
♠❛tr✐③ Pn×n ❡①✐st❡✱ ❡❧❛ é ❞✐t❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ Mn×n ❡ M s❡ ❞✐③ ✐♥✈❡rtí✈❡❧✱
♥❡ss❡ ❝❛s♦ t❡♠♦s P =M−1✳
❆ s❡❣✉✐r ❝❛❧❝✉❧❛r❡♠♦s ❛ ✐♥✈❡rs❛ ❞❛ ♠❛tr✐③M =
1 2 3 4
✳ ❖✉ s❡❥❛✱ ❞❡✈❡♠♦s
❡♥❝♦♥tr❛r ✉♠❛ ♠❛tr✐③ P =
a b c d
✱ t❛❧ q✉❡✿
1 2 3 4
| {z }
M
a b c d
| {z }
P = 1 0 0 1 ❖✉ s❡❥❛✱
1a+ 2c= 1 1b+ 2d= 0 3a+ 4c= 0 3b+ 4d= 1
❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛✱ t❡♠♦s q✉❡✿ a=−2✱b = 1✱ c= 3 2 ❡ d=
−1
2 ✳
❈♦♠♦ ♦ s✐st❡♠❛ ❢♦✐ ♣♦ssí✈❡❧ ❡ ❞❡t❡r♠✐♥❛❞♦✱ ❛ ♠❛tr✐③M é ✐♥✈❡rtí✈❡❧ ❡ ♣♦ss✉✐ ✐♥✈❡rs❛ P✱ ❞❛❞❛ ♣♦r✿
P =
−2 1
3/2 −1/2
❖❜s❡r✈❡ q✉❡ M P =P M =I2✳
❆❣♦r❛ ✈❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ♥♦ q✉❛❧ M ♥ã♦ é ✐♥✈❡rtí✈❡❧✳ ❈♦♥s✐❞❡r❡ M =
1 2 2 4
✱ ♣❛r❛ ✐ss♦ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ♠❛tr✐③ P =
a b c d
✱ t❛❧ q✉❡✿
1 2 2 4
| {z }
M
a b c d
| {z }
P
=
1 0 0 1
❖✉ s❡❥❛✱
1a+ 2c= 1 1b+ 2d= 0 2a+ 4c= 0 2b+ 4d= 1
❖ s✐st❡♠❛ ❛❝✐♠❛ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦✱ ♣♦✐s ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ♣♦r 2 ❡ ❝♦♠♣❛r❛♥❞♦✲❛ ❝♦♠ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✱ ❝♦♥❝❧✉✐r❡♠♦s q✉❡0 = 2✱ q✉❡ ♥ã♦ é ✈❡r❞❛❞❡✐r♦✳ ▲♦❣♦ ❛ ♠❛tr✐③P ♥ã♦ ❡①✐st❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱M ♥ã♦ é ✐♥✈❡rtí✈❡❧✳
✸✳✷✳✸ ▼❛tr✐③ ❞❡ r♦t❛çã♦
❱❛♠♦s ❞❡✜♥✐r ❛❣♦r❛ ✉♠❛ ♠❛tr✐③ ❝❤❛♠❛❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡ ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ♦ ✈❡t♦r OP~ ✱ ♥ã♦✲♥✉❧♦✱ s❡♥❞♦ O ❛ ♦r✐❣❡♠ ❞❡ss❡ ♣❧❛♥♦ ❡ P = (x, y)✳
❙✉♣♦♥❤❛ q✉❡ ♦ â♥❣✉❧♦ ❡♥tr❡OP~ ❡ ♦ ❡✐①♦xs❡❥❛α✳ ❆♦ r♦t❛❝✐♦♥❛r ♦ ✈❡t♦rOP~ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦ ❞❡ ✉♠ â♥❣✉❧♦ θ✱ t❡r❡♠♦s ♦ ✈❡t♦r OP~ ′✱ ❝♦♠P′
= (x′
, y′
)✱ ✜❣✉r❛ ✷✵✳ ❆ s❡❣✉✐r✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✾❪✱ ❡♥❝♦♥tr❛r❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦rOP~ ′✳
❋✐❣✉r❛ ✷✵✿ ❘♦t❛çã♦ ❞♦ ♣♦♥t♦ P
❉♦ tr✐â♥❣✉❧♦ OP Q✱ t❡♠♦s q✉❡✿
✭✷✮ cosα = x
kOP~ k
✭✸✮ sinα = y
kOP~ k
❋❛③❡♥❞♦ ❛ r♦t❛çã♦ ❞❡ OP~ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✱ ❞❡ ✉♠ â♥❣✉❧♦ θ ✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✷✵✱ ♦❜t❡r❡♠♦s ♦ ✈❡t♦r OP~ ′✳ ❉❛í ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿
✭✹✮ cos (α+θ) = x
′
kOP~ ′
k
✭✺✮ sin (α+θ) = y
′
kOP~ ′
k
▲❡♠❜r❛♥❞♦ q✉❡✱ ♣❡❧❛ ❛❞✐çã♦ ❡ s✉❜tr❛çã♦ ❞❡ ❛r❝♦s✱ t❡♠♦s q✉❡✿
cos (α+θ) = cosαcosθ−sinαsinθ
❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s (2)✱ (3) ❡ (4)✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❢❛t♦ ❞❡ kOP~ ′
k=kOP~ k✱ s❡❣✉❡ q✉❡✱
✭✻✮ x′
=xcosθ−ysinθ
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ❝♦♠♦✱
sin (α+θ) = sinαcosθ+ sinθcosα
❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s (2)✱ (3) ❡ (5)✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❢❛t♦ ❞❡ kOP~ ′
k=kOP~ k✱ s❡❣✉❡ q✉❡✱
✭✼✮ y′
=ycosθ+xsinθ
❚❡♠♦s ♣♦rt❛♥t♦ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✿
x′
=xcosθ−ysinθ y′
=ycosθ+xsinθ
❊ss❡ s✐st❡♠❛ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♣♦r ♠❡✐♦ ❞❛ ♥♦t❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱
x′ y′ =
cosθ −sinθ sinθ cosθ
x y
❖✉ ❛✐♥❞❛ ♥❛ ❢♦r♠❛✿
P′
=M P
❙❡♥❞♦ P′ = x′ y′
, M =
cosθ −sinθ sinθ cosθ
❡ P =
❆ ♠❛tr✐③ M é ❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❞♦ â♥❣✉❧♦ θ ❞❡ ♦r❞❡♠ 2✳
❆ s❡❣✉✐r✱ ❡♥❝♦♥tr❛r❡♠♦s(x′
, y′
)✱ ♦r✐❣✐♥ár✐♦s ❞❛ r♦t❛çã♦ ❞❡OP~ ❡♠90◦✳ ❙❡♥❞♦
P = (1,2)✳ ◆❡st❡ ❝❛s♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ✈❡t♦r✲❝♦❧✉♥❛✿
P′
=M P =
cos 90♦ −sin 90♦ sin 90♦ cos 90♦
1 2
❖✉ s❡❥❛✱
P′
=
0 −1
1 0
1 2
=
−2
1
❆ ✜❣✉r❛ ✷✶ ❛♣r❡s❡♥t❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ✈❡t♦r❡s OP~ ❡ OP~ ′✳
❋✐❣✉r❛ ✷✶✿ ❘♦t❛çã♦ ❞♦ ✈❡t♦rOP~ = (1,2) ♣❛r❛ OP~ ′ = (−2,1)
❙❡❣✉♥❞♦ ❬✶✵❪✱ ♣❛r❛ r♦t❛❝✐♦♥❛r ❡♠ R3✱ é ♥❡❝❡ssár✐♦ ❡s♣❡❝✐✜❝❛r ✉♠❛ r❡t❛ q✉❡✱ ❡♠ t♦r♥♦
❞❛ q✉❛❧✱ s❡ ❞❛rá ❛ r♦t❛çã♦✳
❆ r♦t❛çã♦ ♣♦❞❡ s❡ ❞❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ t♦r♥♦ ❞❡ ❝❛❞❛ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦✳ ❖ s❡♥t✐❞♦ ❞❡ r♦t❛çã♦ s❡❣✉❡ ❛ r❡❣r❛ ❞❛ ♠ã♦ ❞✐r❡✐t❛✺✱ ✜❣✉r❛ ✷✷✳
✺❆ r❡❣r❛ ❞❛ ♠ã♦ ❞✐r❡✐t❛ ✐♥❞✐❝❛ q✉❡✱ ❝♦♠ ♦ ❞❡❞♦ ♣♦❧❡❣❛r ❞❛ ♠ã♦ ❞✐r❡✐t❛ ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❞♦ ❡✐①♦✱ ❛ r♦t❛çã♦ s❡ ❞❛rá ♥❛ ❞✐r❡çã♦ ❞♦s ❞❡♠❛✐s ❞❡❞♦s ❞❛ ♠ã♦✳
❋✐❣✉r❛ ✷✷✿ ❘♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ x✱ ❡✐①♦ y ❡ ❡✐①♦ z✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ z é ❛ ♠❡s♠❛ ❡①♣r❡ssã♦ ❞❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❞❡ ♦r❞❡♠2✱ ✈✐st❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛❝r❡s❝✐❞❛ ❛♣❡♥❛s ❞❛ t❡r❝❡✐r❛ ❧✐♥❤❛✳
x′
=xcosθ−ysinθ y′
=xsinθ+ycosθ z′
=z
❊ss❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❝♦♠♦✿
x′
y′
z′
=
cosθ −sinθ 0 sinθ cosθ 0
0 0 1
x y z
❱❡❥❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ r♦t❛çã♦ ❞❡ ✉♠ ✈❡t♦r V~ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ z✱ ❞❡ ✉♠ â♥❣✉❧♦θ✳
❋✐❣✉r❛ ✷✸✿ ❱❡t♦r ❣✐r❛♥❞♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ z
❏á ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ x✱ ❛s ❝♦♦r❞❡♥❛❞❛sx′✱
y′ ❡
z′✱ sã♦ ❞❛❞❛s ♣♦r✿
x′
=x y′
=ycosθ−zsinθ z′
=ysinθ+zcosθ
❊ss❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❝♦♠♦✿
x′ y′ z′ =
1 0 0
0 cosθ −sinθ 0 sinθ cosθ
x y z
❱❡❥❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ r♦t❛çã♦ ❞❡ ✉♠ ✈❡t♦r V~ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ x✱ ❞❡ ✉♠ â♥❣✉❧♦θ✳
❋✐❣✉r❛ ✷✹✿ ❱❡t♦r ❣✐r❛♥❞♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ x
❊♠ t♦r♥♦ ❞♦ ❡✐①♦ y✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♦r❞❡♥❛❞❛s ❞❡ x′✱
y′ ❡
z′✿
x′
=zsinθ+xcosθ y′
=y z′
=zcosθ−xsinθ
❊ss❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❝♦♠♦✿
x′ y′ z′ =
cosθ 0 sinθ
0 1 0
−sinθ 0 cosθ
❱❡❥❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ r♦t❛çã♦ ❞❡ ✉♠ ✈❡t♦r V~ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ y✱ ❞❡ ✉♠ â♥❣✉❧♦θ✳
❋✐❣✉r❛ ✷✺✿ ❱❡t♦r ❣✐r❛♥❞♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ y
P❛r❛ t♦❞♦s ♦s três ❝❛s♦s ❛♥t❡r✐♦r❡s✱ ❛ r❡♣r❡s❡♥t❛çã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❢❡✐t❛ ❝♦♠♦✿
P′
=RθP
❙❡♥❞♦ Rθ ❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦ θ✳
❱❡r❡♠♦s ❛❣♦r❛ ❝♦♠♦ ❢❛③❡r ♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ✈❡t♦r✲❧✐♥❤❛ ♣♦r ✉♠❛ ♠❛tr✐③✳
❈♦♥s✐❞❡r❡ ✉♠❛ ♠❛tr✐③ A =
a11 a12
a21 a22
✱ ❛❣♦r❛ r❡❡s❝r❡✈❛✲❛ ❝♦♠♦ A =
~u ~v
✱ s❡♥❞♦ ~u = a11 a12
❡~v = a21 a22
✳ ▼✉❧t✐♣❧✐❝❛r ✉♠❛ ♠❛tr✐③ ❧✐♥❤❛ a b ♣❡❧❛ ♠❛tr✐③ A✱ é ❝♦♥s✐❞❡r❛r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦ ✈❡t♦r a~u+b~v✱ ♦✉ s❡❥❛✱ a
~u 0
+b
0 ~v
✳
❆ ✜❣✉r❛ ❛ s❡❣✉✐r ❛♣r❡s❡♥t❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✿
(2,3)
5 −4
6 7
= (28,13)
❋✐❣✉r❛ ✷✻✿ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠❛ ♠❛tr✐③
❆ s❡❣✉✐r tr❛t❛r❡♠♦s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ 2✳
2 3
−2 1
~u ~v
=
2~u+ 3~v −2~u+ 1~v
❈♦♥s✐❞❡r❛♥❞♦ ~u= (5,−4) ❡~v = (6,7)✱ t❡♠♦s q✉❡✱
2 3
−2 1
5 −4
6 7
=
28 13
−4 15
❆ ✜❣✉r❛ ❛ s❡❣✉✐r✱ r❡♣r❡s❡♥t❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ♠✉❧t✐♣❧✐❝❛çã♦✳
❋✐❣✉r❛ ✷✼✿ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦ ♣r♦❞✉t♦ ❡♥tr❡ ♠❛tr✐③❡s
◆♦t❡ q✉❡ ♦ ✈❡t♦r q✉❡ r❡s✉❧t❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ♠❛tr✐③
5 −4
6 7
♣❡❧❛ ♣r✐♠❡✐r❛ ♠❛tr✐③ ❧✐♥❤❛ (2,3) ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ é ❛ ❞✐❛❣♦♥❛❧ ❞♦ r❡tâ♥❣✉❧♦ ❝✉❥♦s ❧❛❞♦s é ✐❣✉❛❧ ❛ 2(5,−4) ❡ 3(6,7)✳ ❆ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛ ♥♦s ❛❥✉❞❛ ❛ ✈✐s✉❛❧✐③❛r✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ ♦ ♦✉tr♦ ✈❡t♦r✱ r❡s✉❧t❛♥t❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ s❡❣✉♥❞❛ ♠❛tr✐③ ❧✐♥❤❛ (−2,1) ♣❡❧❛ ♠❛tr✐③ ♦r✐❣✐♥❛❧✳
✸✳✸ ❉❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ s❡❣✉♥❞❛ ♦r❞❡♠
❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✶✶❪✱ ❡ t❛♠❜é♠ ❥á ✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡ ♥❡st❡ tr❛❜❛❧❤♦✱ ✉♠❛ ♠❛tr✐③ q✉❛✲ ❞r❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❡t❡r♠✐♥❛ ✈❡t♦r❡s✲❧✐♥❤❛ ❡ ❡st❡s ❞❡❧✐♠✐t❛♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❛♥❞♦ ❡st❡s ✈❡t♦r❡s✲❧✐♥❤❛ ♥ã♦ sã♦ ♠ú❧t✐♣❧♦s✳
❆ ár❡❛ ❞❡ss❡ ♣❛r❛❧❡❧♦❣r❛♠♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ❛ ♣❛rt✐r ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③✱ q✉❡ é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱
detA=
a11 a12
a21 a22
= a11a22−a21a21
❱❡r❡♠♦s ❛ s❡❣✉✐r q✉❡ ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r A é ♦ ♠ó❞✉❧♦ ❞♦ ❞❡✲ t❡r♠✐♥❛♥t❡ ❞❡A✳
✸✳✸✳✶ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ 2 ❈♦♥s✐❞❡r❡ ❛ ♠❛tr✐③✳
M =
a11 a12
a21 a22
❊ ❛❞♠✐t❛ q✉❡ s❡✉s ✈❡t♦r❡s✲❧✐♥❤❛ ♥ã♦ sã♦ ♠ú❧t✐♣❧♦s✳
❖ ♣❛r❛❧❡❧♦❣r❛♠♦ ❣❡r❛❞♦ ♣❡❧♦s ✈❡t♦r❡s✲❧✐♥❤❛ ❞❡ M✱ ~v = (a11, a12) ❡ w~ = (a21, a22)✱ ❡stá
❛♣r❡s❡♥t❛❞♦ ❛ s❡❣✉✐r✳
❋✐❣✉r❛ ✷✽✿ P❛r❛❧❡❧♦❣r❛♠♦
