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Faculdade Esta´cio do Recife CA´LCULO VETORIAL E GEOMETRIA ANALI´TICA Prof. Se´rgio Barreto L I S T A D E E X E R C I´ C I O S - V E T O R E S 2 1. Encontre o produto interno −→ u • −→v nos casos abaixo: (a) −→ u = ( 2, 1 ) e −→ v = ( −4, 2 ) (b) −→ u = ( −1, 4 ) e −→ v = (√ 3, √ 2 ) (c) −→ u = ( −2, 0 ) e −→ v = ( 3,−1 ) (d) −→ u = ( 4, 3 ) e −→ v = ( 4, 5 ) (e) −→ u = ( 4, 7, 3 ) e −→ v = ( 2, 1, 1 ) (f) −→ u = ( 0,−5, 2 ) e −→ v = ( 3, 2,−6 ) (g) −→ u = ( 7, 1,−7 ) e −→ v = ( 0, 1,−1 ) (h) −→ u = ( −2,−1, 3 ) e −→ v = ( −4, 3,−1 ) 2. Encontre a norma do vetor −→ v nos casos abaixo: (a) −→ v = ( −4, 1 ) (b) −→ v = (√ 3, √ 2 ) (c) −→ v = ( 3, 0 ) (d) −→ v = ( 4, 4 ) (e) −→ v = ( 7, 2, 1 ) (f) −→ v = ( 3,−5,−6 ) (g) −→ v = ( 7, 7,−7 ) (h) −→ v = ( −4, 3,−1 ) 3. Ate´ que ponto deve-se prolongar o segmento de extremos A = (2, 0, 0) e B = (0, 2, 0) , no sentido de A para B, de modo que seu comprimento quadruplique? 4. Dado −→ v = ( a,−2 ) , calcular os valores de a para que se tenha ‖−→v ‖= 3. 1 5. Verifique quais dos vetores abaixo e´ unita´rio. (a) −→ v = ( 1 2 , 1 2 ) (b) −→ v = ( 1 3 , 2 3 ) (c) −→ v = (√ 3 2 , 1 2 ) (d) −→ v = ( 3 5 ,−4 5 ) (e) −→ v = (√ 3 2 , 1 2 , 1 2 ) (f) −→ v = (√ 3 2 , 1 2 , 0 ) (g) −→ v = (√ 5 4 , 1 4 , √ 10 4 ) (h) −→ v = ( 1 3 , 2 3 ,−1 ) 6. Encontre os versores dos vetores abaixo: (a) −→ v = ( −4, 1 ) (b) −→ v = (√ 3, √ 2 ) (c) −→ v = ( 3, 0 ) (d) −→ v = ( 4, 4 ) (e) −→ v = ( 7, 2, 1 ) (f) −→ v = ( 3,−5,−6 ) (g) −→ v = ( 7, 7,−7 ) (h) −→ v = ( −4, 3,−1 ) 7. Encontre o aˆngulo entre os vetores −→ u e −→ u nos casos: (a) −→ u = ( 2, 1 ) e −→ v = ( 4, 6 ) (b) −→ u = ( 7, 5 ) e −→ v = ( 1, 2 ) (c) −→ u = ( −2, 0 ) e −→ v = ( 3,−1 ) (d) −→ u = ( 4, 3 ) e −→ v = ( 4, 5 ) (e) −→ u = ( 2, 1, 4 ) e −→ v = ( 4, 6,−5 ) (f) −→ u = ( −3, 1,−2 ) e −→ v = ( 0, 1,−1 ) (g) −→ u = ( 1, 1,−3 ) e −→ v = ( 1,−1, 2 ) (h) −→ u = ( −2,−1, 3 ) e −→ v = ( −4, 3,−1 ) 2 8. Encontre −→ u × −→v nos casos abaixo: (a) −→ u = ( 2,−1, 3 ) e −→ v = ( 4, 1, 1 ) (b) −→ u = ( 0, 2, 0 ) e −→ v = ( 1, 3,−1 ) (c) −→ u = ( 1, 1, 2 ) e −→ v = ( 3, 1,−1 ) (d) −→ u = ( 2, 1, 0 ) e −→ v = ( 0, 1,−1 ) (e) −→ u = ( 2, 1, 4 ) e −→ v = ( 4, 6,−5 ) (f) −→ u = ( −3, 1,−2 ) e −→ v = ( 0, 1,−1 ) (g) −→ u = ( 1, 1,−3 ) e −→ v = ( 1,−1, 2 ) (h) −→ u = ( −2,−1, 3 ) e −→ v = ( −4, 3,−1 ) 9. Calcular, se poss´ıvel, as a´reas do paralelogramo e do triaˆngulo determinados pelos vetores −→ u e −→ v , nos casos abaixo: (a) −→ u = ( 2,−1, 3 ) e −→ v = ( 4, 1, 1 ) (b) −→ u = ( 0, 2, 0 ) e −→ v = ( 1, 3,−1 ) (c) −→ u = ( 1, 1, 2 ) e −→ v = ( 3, 1,−1 ) (d) −→ u = ( 2, 1, 4 ) e −→ v = ( 4, 6,−5 ) (e) −→ u = ( −3, 1,−2 ) e −→ v = ( 0, 1,−1 ) (f) −→ u = ( 1, 1,−3 ) e −→ v = ( 1,−1, 2 ) 10. Encontre (−→ u , −→ v , −→ w ) , produto misto dos vetores −→ u , −→ v e −→ w , nos casos abaixo: (a) −→ u = ( 0, 2, 1 ) , −→ v = ( 1, 3, 4 ) e −→ w = (−1, 4, 2). (b) −→ u = ( 1, 1, 3 ) , −→ v = ( 2,−1, 5) e −→w = (4,−3, 1). (c) −→ u = ( 2, 4,−5), −→v = (7,−1, 0) e −→w = (2, 2, 3). 11. Verificar se A, B, C e D sa˜o pontos coplanares nos casos abaixo: (a) A = ( 1, 1, 0 ) , B = ( 0, 2, 3 ) , C = ( 2, 0,−1) e D = (−1, 3, 5). (b) A = ( 2, 3, 4 ) , B = ( 1,−1, 9), C = (5,−3, 7) e D = (0, 3, 6). (c) A = ( 1, 1, 1 ) , B = ( 1, 2, 1 ) , C = ( 3, 0, 1 ) e D = ( 5, 7, 10 ) . 12. Calcular, se poss´ıvel, os volumes do paralelep´ıpedo e do tetraedro determinados pelos vetores −→ u , −→ v e −→ w , nos casos abaixo: (a) −→ u = ( 0, 2, 1 ) , −→ v = ( 1, 3, 4 ) e −→ w = (−1, 4, 2). (b) −→ u = ( 1, 1, 3 ) , −→ v = ( 2,−1, 5) e −→w = (4,−3, 1). (c) −→ u = ( 2, 4,−5), −→v = (7,−1, 0) e −→w = (2, 2, 3). 3