Lista 2 Produto Escalar com Respostas

Centro Universitário UNA\nGeometria Analítica - Produto Escalar\n\n1. Dados os vetores u = (2, -3, -1) e v = (1, -1, 4), calcule:\n(a) 2u . (-v)\n(b) (u + 3v) . (v - 2u)\n(c) (u + v) . (u - v)\n(d) (u . u)\n\n2. Sejam os vetores u = (2, -2, 1), x = (3, 1, -2) e w = (2n - 1, -2, 4). Determine a de modo que u = (x + y + z)\n\n3. Dados os pontos A(4, 0, -1), B(2, 0, -1) e C(1, 3, 2) os vetores u = (2, 1, 1) e v = (-1, -2, 3)\n\n4. Determine o vetor w paralelo ao vetor u = (2, -1, -3), tal que w = i - 42.\n\n5. Dados os vetores u = (1, 2, -3), v = (2, 0, -1) e x = (3, 1, 0), determine o vetor z tal que z = (4, 7, 2).\n\n6. Sabendo que |u| = 2, |v| = 3 e u . v = -1, calcule:\n(a) (u . v)\n(b) (2u - v). (2v)\n(c) (u + v) . (6 - 4u)\n(d) (3u + 4u). (-2u - 5v)\n\n7. Os pontos A, B e C são vértices de um triângulo equilátero cujo lado mede 20 cm. Calcule AB . AC e AB . CA\n\n8. Qual é . x . α de para que os vetores u = αr + 2y - 4z e v = -2z + (1 - 2α)y + 3k sejam ortogonais?\n\n9. Dados os vetores u = (2, 1, 0) e v = (x - 2, 5, 2), determine o valor de x para que o vetor u + v seja ortogonal ao vetor z - 2.\n\n10. Dados os pontos A=(-1, 0, 5), B=(-2, -1, 4) e C(1, 1, 1), determine x tal que AC e BC sejam ortogonais, sendo P(x, 7, x - 3). 11. Determine o vetor u tal que |u| = 2, o ângulo entre u e v é 45º e u é ortogonal a v = (1, 1, 0).\n\n12. Seja o vetor x = (2, -1, 1). Obtenha:\n(a) um vetor ortogonal a x.\n(b) um vetor unitário ortogonal a x.\n(c) um vetor de módulo 4 ortogonal a x.\n\n13. Determine os ângulos entre os vetores:\n(a) x = (2, -1, -1) e y = (-1, -1, 2)\n(b) x = (1, -2, -1) e y = (-1, 1, 0)\n\n14. Calcule k para que o ângulo de 30º entre os vetores x = (-3, 1, n) e k é k.\n\n15. Determine o valor de k para que os vetores x = (-2, 3) e v = (k, -4) sejam:\n(a) paralelos.\n(b) ortogonais.\n\n16. Obter os dois vetores unitários ortogonais a cada um dos vetores:\n(a) 4i + 3j\n(b) -2, 2, 3\n\n17. Determine o valor de k para que seja 45º o ângulo entre os vetores u = (2, 1) e v = (1, α).\n\nRespostas\n1) a) -2 b) 21 c) -4 i) 4 2) a) \\frac{2}{3} b) (3, 6, -9) 3) a) (3, 6, -9) b) (-4, -\\frac{3}{2}, 1) 4) (-6, 3, -9) 5) x = (2, -3, 4) 6) y = (2, -3, 4) a) 7 b) 38 c) -4 d) -181 7) 200 e -200 8) 5 9) 3 ou -6 10) x = -5\ninfinito positivos: (1,1,1) b) Um deles: (\\frac{-3}{3}, \\frac{-7}{3}) c) Um deles: (\\frac{-3}{3}, \\frac{-2}{3}) \n\n13) a) 120º b) 150º c) (\\frac{-3}{\\sqrt{3}}, \\frac{3}{\\sqrt{3}})\n17) 3 ou -\\frac{1}{2}\n Lista 2\n\n1) a) 2u . (-v)\n2(2, -3, -1) . (-3, -4, 1)\n= 2*2+(-3)*(-4)+(-1)*1\n= 4 + 12 - 1\n= +15\nTot = -2\n\nb) (u + 3v) . (v - 2u)\n(2, -3, -1) + (3*4, 3*4, -3*3)\n= (2 - 3*4, -3 + 3*4, -1 + 3*(-3))\n= (-10, 9, -10) . (4, -3, -1)\n= -10*4 + 9*(-3) + (-10)(-1)\n= -40 - 27 + 10\n\n= -57\nTot = 3\n\nc) (u + v) . (u - v)\n(2, -3, -1) + (1, -4, 1)\n= (3, -7, 0)\n(2, -3, -1) - (1, -4, 1)\n= (1, 1, -2)\n= 15 - 20 + 66\nTot = -4\n\nd) ((2, -3, -1) + (1, -4, 4)). ((1, -2, 4) - (2, -3, -1))\n= (-3, -4, 2) . (-1, 1, 4)\n= -3*(-1) + (-4)*5\nTot = 4\n u^=0(2,a,-1)\n(3,-1,-2)\n6+a+2\nA-6+2\nA+B\n\na)(u+v)\n(2,a,-1)+(3,i,-2)-(5,a,1,-3)\nv+w=(3,1,-2)+(2a-1,-2,4)=>(2a+2,-1,2)\n\n(u.v).(5,a,1,-3)(2a+2,-1,2)\n10a+0.a-1-6\n10a-a+10-1-6\n=9a+8!\n\nu.v=a+B=9a+3\n\n8a=9-3=5\n8a=5\na=5\n\na=5\nx=(3,6,-9) (a,b,c).x=(u.v).x=-(u.v)(3x)\n((-3,5,1).(-1,-2,3))x=[(2,1,1)(-1,-2,3)]\n(-1,10+3)x=-(2.2+3)\n-6x+(-1.(-1.2,3))-3x\n6x+3x=(1,2,3)\nX=(1/3,-2/3,3/3)->X=(-1/3,-2/3,1)\n\nX.u.v=-42\n(2x-x,3)&(2,-1,3)\n4x+x+9x=-42\n\n14x=-3\nX=-3\nX.v=x.v=0\nx.u.v=(a,b,c).(1,2,0)\n2a-6=0\n\nx+2b+3c=-16\na+2b-3(2a)-d.6\na+2b.6a=-d6\n-5a+2b>1d6 a+2b-36=-16\n2a-c=0\n3a+b=3\n\n6a+2b=6\n-5a+2b=-16\n\n2a-c=0\nc=2a\nc=2.2\n\nx=(2,-3,4)\n\n\na)(u-3,v).u\n2.2-3(1-1)\n4+3\neu.v=7\n\nb)(2u.v).2(v)\n4.3-3.2-(1)\n36+2\n\nc)(-A+i.v).(v-4u)\n-1-4.2.2+3.3-1\n-1-1.6+0+9\n=-4\n\nd)(3u+4v).(-2u-5v)\n-6u.v-8u.v-15u.v-20u.v**\n-6(1-2.3u.v-2|v|)\n-6.2^2.3(1)-20.3^2\n-24+23-80\n\n198 1)\n\\bar{AB}, \\bar{AC}, \\bar{CA}\n\\bar{AB}.\\bar{AC} = [\\bar{AB}, [\\bar{AC}].\\sin 60^\\circ\n\\bar{AB} = 20\n\\bar{AB} = \\bar{AC} = 200\n\\Rightarrow \\bar{AB} - \\bar{AC} = 200\n\\bar{AB}.\\bar{AC} = 200\n\n8) a = x^2 + 2y^2 - 4z = b = 2, 1(d, 2a)\\frac{7}{2}k \\text{ ortogonales.\n\\bar{a} = a - 1 + 6 = 0\na(1 + 1)(2).2+4 = 0\na(2 + 1)(4,4) = 0\na^2 + a + 0 + 6 = 0\na^2 + b + 0 + 6 = 0\nC - 6 = 0\n2a^3 + 6a - 36 = 0\na^2 + 3a - 18 = 0\na - 3)(a + 6) = 0 10\n\\bar{AC} \\in \\overline{BP} \\text{ orthogonal } P(x, 0, x-3)\n\\bar{AC} = (c, -a) = (d, d, 0) - (-1, 0, 5)\n\\bar{AC} = (2, 1, 4)\n\n\\bar{BP} = \\bar{B} - \\bar{P} = (x_0, x-3) - (2_1, 1, 4)\n\\bar{BP} = x - 2, jx - 7\n\n\\bar{AC} . \\bar{BV} = 0\n(2, 1, 4).(x-2, 1, x-7) = 0\n2(x-2) + 1 - 4(x-7) = 0\n2x - 4 + 1 + 28 = 0\n-2x = -25\n\\Rightarrow x = \\frac{25}{2}\n\n11\\text{ Determinar } \\bar{u} \\text{ tal que } ||\\bar{u}|| = 2\n\\bar{u} = (x_1, y_1, z_1).(1, -1, 0)\n0 = -x_1 + y_1 + 2.0\n0 = -x + y = 0\nu . \\bar{u} = ||\\bar{u}||\\cos \\theta \\text{ so } 2\n(x_1, y_1, z_1)(1, -1, 0) = \\alpha\n\\text{ } x - y = 2\n\nx + y = 0\nx - y = -2\n2y = 2 \\Rightarrow y = 1\nx = -1\n\n 12\\text{ conjunto } \\bar{J} (2, -1, 1) \\text{ Concatenate }:\na) \\bar{u}, \\bar{v} \\text{ ortogonales } \\bar{u}, \\bar{v} = 0\\text{ (donde sea } = 0)\n(x, y, z)(-1, -1, 1) = 0\n2x - y - z = 0\n2 = y - 2x \\Rightarrow\\bar{v}_{i} = 1, (-1, 1)\n\nb) \\text{ Unitario ortogonal } \\bar{u} = (1, -1, -\\sqrt{3})\n= -\\left(\\frac{1}{\\sqrt{3}}, \\frac{-1}{\\sqrt{3}}, \\frac{-1}{\\sqrt{3}}\\right)\n\\text{ Miembro de } \\bar{u} \\text{ viene }\n\nc) \\text{ Valu propio de } \\bar{u}\\text{ and }\\bar{v}.\n\\bar{u} = \\frac{1}{\\sqrt{3}}(1, -1, -1)\n\\bar{v} = \\left(\\frac{4}{\\sqrt{3}}, \\frac{4}{\\sqrt{3}}, -\\frac{4}{\\sqrt{3}}\\right)\\text{ Modo.}\n\n14\\text{ n prasa feja 300. Llev. Vel }(-3, 1, n \\in \\mathbb{R}^{2})\n\\bar{y} = (-2^3, 1, n)\nK(0, 0, k)\n|v.k| = |j|0 + 0 + n| = n_y\n|v| = \\sqrt{3}^2 + 1^2 + n^2 \\Rightarrow \n\\sqrt{10n^2}\n|k| = \\sqrt{1} = j Aos 30° = \\( \\frac{n}{\\sqrt{10+n^2}} \\cdot d \\)\\n\\( \\frac{\\sqrt{3}}{2} = \\frac{n}{\\sqrt{10+n^2}} \\)\\n3n^2 = 3 + 3n^2\\nn = -\\sqrt{30} \\slash\\n\\n(15) Bestimmte K prand vektor (\\( u = (-2,3) \\),\\( v = \\sqrt{(k,-4)}\\)\\na) Pranlador\\n\\( u = x \\cdot v \\)\\n(-2,3)\\n\\(-2,3)(k-4)\\n -2k + 12 = 0\\n-2k = -12\\n\\{ k = -6\\}...\\nb) erlagoml\\n\\( u.v = 0\\)\\n(-2,3)(k-4) = 0\\n -2k + 12 = 0\\n-2k = 12\\n\\{ k = 12 \\} \\n\\{ k = \\frac{8}{3} \\} \\n\\{ k = \\frac{4}{3}\\} 1) Setze Werte a, p/4se Long Vectors \\( u = (-2,1) \\) e \\( v = (1,a)\\) \\n\\( |u| = \\sqrt{x^2+y^2} = \\sqrt{2 + a^2} \\)\\n\\( + |u| = \\sqrt{2 + a^2} \\)\\n\\( |v| = \\sqrt{1+a^2} \\)\\n\\( |u| = \\sqrt{v^2 + a^2}\\)\\n\\( cos\\theta = \\frac{4s}{2} \\)\\n(2 + a^2) = \\( (\\sqrt{5} \\cdot \\sqrt{1+a^2} \\cdot \\frac{\\sqrt{2}}{2})^2\\)\\n4x + a^2 - 10 - 4y = 0\\na^2 - 2x - (4 + 10 - a + a^2) = 0\\n\\( 2a^2 - 4 - 10 - d - d = 0 \\)\\n2a - 16 = 0\\na = 8\\na = 4\\n\\( \\Delta = b^2 - 4ac\\)\\n\\( \\Delta = 26 - 4 \\cdot 2.0\\)\\n\\( \\Delta = 16 - 8 \\)\\n\\( \\Delta = 8 \\)