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Geometria Analítica
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5. lección.\n1a. √(j1,0), √(0,-3,2), √(-1,0,1), √(10,-1), u(j1,-2,3)\n\n1\n\n1 x + O + 2\n\n-2 -x 2y n2\n\n-3 O + d\n\nAd = | 0 1 0 |\n\n | -3 -4 2 | d = (12 - 3) - (14)\n\n | 0 2 -4 | Ad = 0\n\n\njagando o para, termoz\n\nb. 6 1 0 1\n\n13 = x - j + q-1 - q + 4\n\n5 0 2 1 -4\n\n\n(x + 0 + 6 = 6\n\nx - 3y + n2 = 13\n\n0 + 4 - 4 = 5\n\nAd = | 1 0 1 |\n\n | 6 0 1 |\n\n | 6 0 1 |\n\n Ad = 1\n\nAd = d - 0, Adx = 0, Ady = 0, Ad0\n\n0 (1 | - 3 7 13 -3\n\n\n| 1 0 1 |\n\n| | 6 Adx = (t2 + 0 + 2) - (-15 + 84) = -dAdx = 29\n\n bx = 29\n\nAd = | 3 6 1 |\n\n | 1 -6 dAdY = -(5x15 ) - (25- x4)\n\n | 0 5 -9 0 5\n\n 0\n\nAdY = 58\n\nAdz = | 0 0 6 | 0 |\n\n | 1 O | 0 (-15 - 11) - (26)\n\n | 0 3 | -29 = 29\n\n d3 = 9\n\n0 Adg3 \n1 0 0 1 0 \n1 3 9 1 -3 \n0 2 13 0 2 \nAdg2 = (-29) + (4) \n1 0 0 1 = (-43) \n0 0 0 0 = \n0 \n\n- Sistema impossível \na. \n| 0 | 0 | | 1 | \n| 4 | + 0 | 2 | 3 | 7 | 1 | \n| 0 | | | 4 | -4 | \n\nx * 0 + y3 = 12 \nAd1 1 0 1 | Ad0 = 0 \nx -3y3 = 4 \n1 -3 7 \n0 + y2 y3 = -4 \n1 0 3 -4 \n\nAdx 12 0 1 | 12 0 | Ad7 (144 .1) - (12 + 168) \n= -3 -3 = 1 \n-4 y2 = -4 = 0 \nAdvr = -36 \n-d \n\nAdg1 1 12 1 | \n12 Ad1 (-4 -4) - (-28 -48) \n4 1 7 | Adg -68 \n0 -4 0 -4 \n| \n\n- Sistema impossível. \n2a. (1,1); (0,0); (2,4)(6) = 0 \n| 1 | 0 | | 2 | 0 | \na | 1 | + b 0 | + c 2 | = 0 \n| 1 | + b | + c | 4 | = 0 \n| 1 | 0 | 6 | 0 | \n a. 0 + b 0 + a c = 0 \na b + 4 c = 0 \n10 a + b c = 0 \na + 4 b + 6 c = 0 \npara a = 1 termos: para a = 8 termos \n1 + -2 c = 0 + c = -1/2 \n1 + -2 b -4 c = 0 + b 1/2 \n1 + 4 b 6 c = 0 \n= Substituindo: \n1 + 1 - 3 c = 0 \n0 = 0 \n= 0 = 0 \npassível \n\nb (1,0) + b(6,2) + c(6,4,2) = 0 \na + 6 b + 6 c = 0 \n40 a + 6 b - 15 h = 0 \n40a + 6b + c = 0 \n5 b + c = 0 \n= -5 b + c = 0 \n8 a = 2 c = 0 \n1 3 b = \n4 8 \n= 3 - 1 c = 0 \n= 3 - 2 = 0 \n\nparei d 1 termo \n2 1 6 - 5 \n= Jo + 19 10 - 0 \n= 4 - 0 \n9 4 - 3 0 \n c. a(1,1) + b(1,0) + c(0,1) = 0 \na + b + c = 0 \n2 + 7 + b = a - a + b = 0 \na - 4 b = 0 \na - 10 y3 = 0 + 0: 0 \n\npara a = 1 termos: para a = 6 termos: \nb = 4 \n1 = 2 \n2 = 4 \n= substituindo: \n1 + 4b + 6 c = 0 \n1 - 4 x -3 c = 0 \n0 = 0 \npassível \nb(1,1) + b(5,-5) + c(-5,6) = 0 \na + 6 b - 15 c = 0 \n60 a + 6 b = 0 \n+ 60 a + 6 c = 0 \n0 = 3 - 1 ; 4 - 6 \n- 2 + 3 = 6 \nd(6,4)c = d(1,1)(x)=0 \n= (1,2) 0 \n-14 x + 6 - 5 x 0 \n35 35 \n-14 + 105 = -4 x + 10 = 0 \n= 96 c = 0 \n 30. a. (0,0) (1,2,0) (2,4,x) = 0\n a + b + xc = 0\n 0a + 2b + 4c = 0\n a + 2b + xc = 0\n | 1 2 | 1 del: x\n | 0 2 4 | 0\n | 0 0 0 | x = 0 -> x = 0\n 2\n b. (1,1) (1,cos(x))\n a + b = 0\n a + cos(x) = 0\n 1 1 - cos(x) = 1\n a + cos(x) = 0\n 1 (cos(x))\n (x = 1 a; -1)\n c. (x,y) (-x,y)\n ax bx -> x - x - y^c = 0\n ax - bx -> x - x = 0 = 0\n 4 for any value of '\n d. (1,1) (4,2,-1) (3,4,0)\n a + nx + 2c = 0 - 1x + y 4y + 2 - (4n)\n a + b + 4c = 0 1 2 3 2 4 - 6 = 0\n a + b + 0 = 0 1 0 1 x = 6\n 4 6 = 3\n 2 40. a. m(0,8,3) + 66:0 + 0x + 81y + 23 + 6:0\n t(8,1,0)(y,6,0)(8,x + 0,1:0)\n P1: P2: (0,8,2)(5,4,0)\n P1:P2: (0,8,0)\n v. + Gk - 1844 + 4 +\n P1:P2: (0,8,0)\nv = (-3,-184+6)\n ayundando como x + terminos:\n B3: 1 - 88 - 1682 = 6, 8 33; B2 -> z = 81/3\n 4 x = 11 (y = 11)\n P1: (0,-11,82);\n 40 \n P3(0,-3 - 8)\n 4\n 2 Proj P2: (27, 46, 16) -> V - Proj P3: (-23, -184, -64)\n (20, 5, 5)\n (0, 5, 5)\n V - Proj P3: (-492, -966, -236)\n (20, 5, 5) que e perpendicular a y\n Arquib: una vez que parametricé es para x y O 0 en D\n P0(0, -11, 82)\n W = (-33, -181, 63)\n V = (0, 0, -8)\n P1(0,-11,0)\n P2:P1(0, Q - 82)\n Proj P3(-33,-194,64) (0,0,82) w\n ||w||^2 =\n (x^2 + (-33)^2 +(-194)^2)\n \n Proj P1 5.248 => Proj P2 5.218\n (-33,-194,64) (-33,-181,64)\n Proj P1,\n Proj P2 = -5.92,103 - (5,-184,64)\n Proj P1(0.3, 1, 0.8, 0.3) w\n Proj P3 = Proj P4 = (-0.13, -0.98, -0.37) w\n (x,y.z,n)\n b.\n\nM1: (1, 80), (x, y, z): 6:0 -> x = 8, y = 6:0\nM2: (8, 0, 1) (x, y, z) = 0 -> 8x + z = 0\nV: (1, 8, 0) x (3, 0, 0) -> V: -1, -6, 8 + V: (8, -1, 6)\n\nLlevando O para y = 0.\n x = 6\n z = 48\n\nP0(-6, 0, 48)\nP3(-6, 0, 0)\n(P2(-0, 0, -48)\n\nProj P2: (0, 0, -48) - (8, -1, 64), y^3 = -30k2 (8, -1, 64)\n(x^7 + (-1)^ + 60x^2)^3\n4.168\n\nProj P0: -1024, (8, -1, 64), Proj P3: (-5.9, 0, 42.5)\n1387\n\nV - Proj P0: (13, 9, -1, 7, 16, 8)\n\nLlevando O para x = 0.\n z = 0\n\nP0(-6, 0, 18)\nP1(0, 0, 0)\nP0(-6, 0, 18)\n\nProj P1: (6, 0, -9) - (8, -1, 64)^3 = -20k3 (8, -1, 64)\n √8^ + (-1)^ + (6, 4)\n 4161\n\nProj P0: -5.8 + 0.72 + 46.5)\nV - Proj P1(13.8, 1, 72, 1.75)\n\nV - Proj P1(V - Proj P2) = 0 -> (13.8, -1.72, 17.5) -> (13.9, -16, 18)\n\n498.92 ≠ 0\n\n< No es posible actuar como para P2.
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5. lección.\n1a. √(j1,0), √(0,-3,2), √(-1,0,1), √(10,-1), u(j1,-2,3)\n\n1\n\n1 x + O + 2\n\n-2 -x 2y n2\n\n-3 O + d\n\nAd = | 0 1 0 |\n\n | -3 -4 2 | d = (12 - 3) - (14)\n\n | 0 2 -4 | Ad = 0\n\n\njagando o para, termoz\n\nb. 6 1 0 1\n\n13 = x - j + q-1 - q + 4\n\n5 0 2 1 -4\n\n\n(x + 0 + 6 = 6\n\nx - 3y + n2 = 13\n\n0 + 4 - 4 = 5\n\nAd = | 1 0 1 |\n\n | 6 0 1 |\n\n | 6 0 1 |\n\n Ad = 1\n\nAd = d - 0, Adx = 0, Ady = 0, Ad0\n\n0 (1 | - 3 7 13 -3\n\n\n| 1 0 1 |\n\n| | 6 Adx = (t2 + 0 + 2) - (-15 + 84) = -dAdx = 29\n\n bx = 29\n\nAd = | 3 6 1 |\n\n | 1 -6 dAdY = -(5x15 ) - (25- x4)\n\n | 0 5 -9 0 5\n\n 0\n\nAdY = 58\n\nAdz = | 0 0 6 | 0 |\n\n | 1 O | 0 (-15 - 11) - (26)\n\n | 0 3 | -29 = 29\n\n d3 = 9\n\n0 Adg3 \n1 0 0 1 0 \n1 3 9 1 -3 \n0 2 13 0 2 \nAdg2 = (-29) + (4) \n1 0 0 1 = (-43) \n0 0 0 0 = \n0 \n\n- Sistema impossível \na. \n| 0 | 0 | | 1 | \n| 4 | + 0 | 2 | 3 | 7 | 1 | \n| 0 | | | 4 | -4 | \n\nx * 0 + y3 = 12 \nAd1 1 0 1 | Ad0 = 0 \nx -3y3 = 4 \n1 -3 7 \n0 + y2 y3 = -4 \n1 0 3 -4 \n\nAdx 12 0 1 | 12 0 | Ad7 (144 .1) - (12 + 168) \n= -3 -3 = 1 \n-4 y2 = -4 = 0 \nAdvr = -36 \n-d \n\nAdg1 1 12 1 | \n12 Ad1 (-4 -4) - (-28 -48) \n4 1 7 | Adg -68 \n0 -4 0 -4 \n| \n\n- Sistema impossível. \n2a. (1,1); (0,0); (2,4)(6) = 0 \n| 1 | 0 | | 2 | 0 | \na | 1 | + b 0 | + c 2 | = 0 \n| 1 | + b | + c | 4 | = 0 \n| 1 | 0 | 6 | 0 | \n a. 0 + b 0 + a c = 0 \na b + 4 c = 0 \n10 a + b c = 0 \na + 4 b + 6 c = 0 \npara a = 1 termos: para a = 8 termos \n1 + -2 c = 0 + c = -1/2 \n1 + -2 b -4 c = 0 + b 1/2 \n1 + 4 b 6 c = 0 \n= Substituindo: \n1 + 1 - 3 c = 0 \n0 = 0 \n= 0 = 0 \npassível \n\nb (1,0) + b(6,2) + c(6,4,2) = 0 \na + 6 b + 6 c = 0 \n40 a + 6 b - 15 h = 0 \n40a + 6b + c = 0 \n5 b + c = 0 \n= -5 b + c = 0 \n8 a = 2 c = 0 \n1 3 b = \n4 8 \n= 3 - 1 c = 0 \n= 3 - 2 = 0 \n\nparei d 1 termo \n2 1 6 - 5 \n= Jo + 19 10 - 0 \n= 4 - 0 \n9 4 - 3 0 \n c. a(1,1) + b(1,0) + c(0,1) = 0 \na + b + c = 0 \n2 + 7 + b = a - a + b = 0 \na - 4 b = 0 \na - 10 y3 = 0 + 0: 0 \n\npara a = 1 termos: para a = 6 termos: \nb = 4 \n1 = 2 \n2 = 4 \n= substituindo: \n1 + 4b + 6 c = 0 \n1 - 4 x -3 c = 0 \n0 = 0 \npassível \nb(1,1) + b(5,-5) + c(-5,6) = 0 \na + 6 b - 15 c = 0 \n60 a + 6 b = 0 \n+ 60 a + 6 c = 0 \n0 = 3 - 1 ; 4 - 6 \n- 2 + 3 = 6 \nd(6,4)c = d(1,1)(x)=0 \n= (1,2) 0 \n-14 x + 6 - 5 x 0 \n35 35 \n-14 + 105 = -4 x + 10 = 0 \n= 96 c = 0 \n 30. a. (0,0) (1,2,0) (2,4,x) = 0\n a + b + xc = 0\n 0a + 2b + 4c = 0\n a + 2b + xc = 0\n | 1 2 | 1 del: x\n | 0 2 4 | 0\n | 0 0 0 | x = 0 -> x = 0\n 2\n b. (1,1) (1,cos(x))\n a + b = 0\n a + cos(x) = 0\n 1 1 - cos(x) = 1\n a + cos(x) = 0\n 1 (cos(x))\n (x = 1 a; -1)\n c. (x,y) (-x,y)\n ax bx -> x - x - y^c = 0\n ax - bx -> x - x = 0 = 0\n 4 for any value of '\n d. (1,1) (4,2,-1) (3,4,0)\n a + nx + 2c = 0 - 1x + y 4y + 2 - (4n)\n a + b + 4c = 0 1 2 3 2 4 - 6 = 0\n a + b + 0 = 0 1 0 1 x = 6\n 4 6 = 3\n 2 40. a. m(0,8,3) + 66:0 + 0x + 81y + 23 + 6:0\n t(8,1,0)(y,6,0)(8,x + 0,1:0)\n P1: P2: (0,8,2)(5,4,0)\n P1:P2: (0,8,0)\n v. + Gk - 1844 + 4 +\n P1:P2: (0,8,0)\nv = (-3,-184+6)\n ayundando como x + terminos:\n B3: 1 - 88 - 1682 = 6, 8 33; B2 -> z = 81/3\n 4 x = 11 (y = 11)\n P1: (0,-11,82);\n 40 \n P3(0,-3 - 8)\n 4\n 2 Proj P2: (27, 46, 16) -> V - Proj P3: (-23, -184, -64)\n (20, 5, 5)\n (0, 5, 5)\n V - Proj P3: (-492, -966, -236)\n (20, 5, 5) que e perpendicular a y\n Arquib: una vez que parametricé es para x y O 0 en D\n P0(0, -11, 82)\n W = (-33, -181, 63)\n V = (0, 0, -8)\n P1(0,-11,0)\n P2:P1(0, Q - 82)\n Proj P3(-33,-194,64) (0,0,82) w\n ||w||^2 =\n (x^2 + (-33)^2 +(-194)^2)\n \n Proj P1 5.248 => Proj P2 5.218\n (-33,-194,64) (-33,-181,64)\n Proj P1,\n Proj P2 = -5.92,103 - (5,-184,64)\n Proj P1(0.3, 1, 0.8, 0.3) w\n Proj P3 = Proj P4 = (-0.13, -0.98, -0.37) w\n (x,y.z,n)\n b.\n\nM1: (1, 80), (x, y, z): 6:0 -> x = 8, y = 6:0\nM2: (8, 0, 1) (x, y, z) = 0 -> 8x + z = 0\nV: (1, 8, 0) x (3, 0, 0) -> V: -1, -6, 8 + V: (8, -1, 6)\n\nLlevando O para y = 0.\n x = 6\n z = 48\n\nP0(-6, 0, 48)\nP3(-6, 0, 0)\n(P2(-0, 0, -48)\n\nProj P2: (0, 0, -48) - (8, -1, 64), y^3 = -30k2 (8, -1, 64)\n(x^7 + (-1)^ + 60x^2)^3\n4.168\n\nProj P0: -1024, (8, -1, 64), Proj P3: (-5.9, 0, 42.5)\n1387\n\nV - Proj P0: (13, 9, -1, 7, 16, 8)\n\nLlevando O para x = 0.\n z = 0\n\nP0(-6, 0, 18)\nP1(0, 0, 0)\nP0(-6, 0, 18)\n\nProj P1: (6, 0, -9) - (8, -1, 64)^3 = -20k3 (8, -1, 64)\n √8^ + (-1)^ + (6, 4)\n 4161\n\nProj P0: -5.8 + 0.72 + 46.5)\nV - Proj P1(13.8, 1, 72, 1.75)\n\nV - Proj P1(V - Proj P2) = 0 -> (13.8, -1.72, 17.5) -> (13.9, -16, 18)\n\n498.92 ≠ 0\n\n< No es posible actuar como para P2.