Lista 01 Ifvv

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❘❯❘❆▲ ❉❖ ❙❊▼■✲➪❘■❉❖✕❯❋❊❘❙❆
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ◆❆❚❯❘❆■❙✕❉❈❊◆
Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❘♦♥❛❧❞♦ ●♦♠❡s ●❛r❝✐❛
■✲▲■❙❚❆ ❉❊ ❊❳❊❘❈❮❈■❖❙ ❉❊ ■◆❚❘❖❉❯➬➹❖ ➚❙ ❋❯◆➬Õ❊❙ ❉❊ ❱➪❘■❆❙ ❱❆❘■➪❱❊■❙✲■❋❱❱
✶✳ ❈❛❧❝✉❧❡ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞♦s ✈❡t♦r❡s (2, 3) ❡ (1, 5)✳
✷✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (1, 2) ❡ q✉❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à ❞✐r❡çã♦ ❞♦ ✈❡t♦r
(−1, 3)✳
✸✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦✱ ♥❛ ❢♦r♠❛ ✈❡t♦r✐❛❧✱ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (3, −1) ❡ q✉❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ 2x − 3y = 7✳
✹✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (1, 2) ❡ q✉❡ s❡❥❛ ♣❛r❛❧❡❧❛ à ❞✐r❡çã♦ ❞♦ ✈❡t♦r (−1, 1)✳
✺✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ✈❡t♦r✐❛❧ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (1, −1) ❡ q✉❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ 2x+y = 1✳
✻✳ ❉❡t❡r♠✐♥❡ ✉♠ ✈❡t♦r ❝✉❥❛ ❞✐r❡çã♦ s❡❥❛ ♣❛r❛❧❡❧❛ à r❡t❛ 3x + 2y = 2.
✼✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞♦ ♣❧❛♥♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ❞❛❞♦ ❡ q✉❡ s❡❥❛ ❡ q✉❡ s❡❥❛ ♣❡r♣❡♥❞✐❝✉❧❛r à ❞✐r❡çã♦
❞♦ ✈❡t♦r η ❞❛❞♦✳
✭❛✮ (1, 1, 1) ❡ η = (2, 1, 3)❀
✭❜✮ (2, 1, −1) ❡ η = (−2, 1, 2)✳
✽✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ✈❡t♦r✐❛❧ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ❞❛❞♦ ❡ q✉❡ s❡❥❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦
❞❛❞♦✳
✭❛✮ (0, 1, −1) ❡ x + 2y − z = 3❀
✭❜✮ (2, 1, −1) ❡ 2x + y + 3z = 1✳
✾✳ ❙❡❥❛♠ u = (a1 , b1 , c1 ) ❡ v = (a2 , b2 , c2 ) ❞♦✐s ✈❡t♦r❡s ♥♦ R3 ✳ ❉❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ❞❡ u ♣♦r v ✱ q✉❡ s❡ ✐♥❞✐❝❛ ♣♦r u ∧ v ✱ ♣♦r

u∧v =

i a1 a2

j b1 b2

k c1 c2

= (b1 c2 − c1 b2 )i + (a2 c1 − a1 c2 )j + (a1 b2 − a2 b1 )k,

♦♥❞❡ i = (1, 0, 0), j = (0, 1, 0) ❡ k = (0, 0, 1)✳ ❱❡r✐✜q✉❡ q✉❡✿
✭❛✮ u ∧ v = −v ∧ u❀
✭❜✮ u ∧ v é ♦rt♦❣♦♥❛❧ ❛ u ❡ ❛ v ❀
✭❝✮ u ∧ (v + w) = u ∧ v + u ∧ w✱ ♦♥❞❡ w = (a3 , b3 , c3 )❀
✭❞✮ (u + v) ∧ w = u ∧ w + v ∧ w✳
✶✵✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ✈❡t♦r✐❛❧ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (1, 2, −1) ❡ q✉❡ s❡❥❛ ♣❡r♣❡♥❞✐❝✉❧❛r às ❞✐r❡çõ❡s
❞♦s ✈❡t♦r❡s u = (1, 1, 1) ❡ v = (1, −2, 1)✳
✶

✶✶✳ ❈❛❧❝❡❧❡ ❛ ♥♦r♠❛ ❞♦ ✈❡t♦r ❞❛❞♦✳
✭❛✮ u = (1, 2)❀
✭❜✮ u = (0, 1, 2)❀
✭❝✮ v = (2, 1, 3)❀
✭❞✮ v =

1 1
2, 3

✳

✶✷✳ ❙❡❥❛ u = (x1 , x2 , . . . , xn ) ✉♠ ✈❡t♦r ❞♦ Rn (n ≥ 2)✳ ▼♦str❡ q✉❡ u ≥ |xi |, i = 1, 2, . . . , n✳
✶✸✳ ❙❡❥❛♠ u ❡ v ✈❡t♦r❡s q✉❛✐sq✉❡r ❞♦