Lista 3 Produto Vetorial com Respostas

Centro Universitário UNA\nGeometria Analítica - Produto Vetorial\nSe \u03bb=3\u00a0-3\u00a0-2k,\u00a0\n\to\u00a0=2i+4j+k\u00b7w=7i+7k,\u00a0\ndetermino:\n(a)\u00a0u\u00a0\u00d7\u00a0u\n(b)\u00a0(2\u00a0\u00d7\u00a0)(3\u00a0)\n(c)\u00a0(i\u00a0\u00d7\u00a0u)+(u\u00a0\u00d7\u00a0i)\n(d)\u00a0(i\u00a0\u00d7\u00a0v)+(v\u00a0\u00d7\u00a0i)\n(e)\u00a0(i\u00a0\u00d7\u00a0x)\n(f)\u00a0(i\u00a0\u00d7\u00a0y)\n(g)\u00a0x\u00a0\u00d7\u00a0x\n(h)\u00a0i\u00a0\u00d7\u00a0v\n(i)\u00a0(7i+5j)\u00a0\u00d7\u00a0w\n(j)\u00a0(v\u00a0\u00d7\u00a0r)\n(k)\u00a0(x\u00a0\u00d7\u00a0y)\n\n2. Efetue:\n(a)\u00a0v\u00a0\u00d7\u00a0k\n(b)\u00a0i\u00a0\u00d7\u00a0(2i)\n(c)\u00a0(3i)\u00a0\u00d7\u00a0(2k)\n(d)\u00a0(i\u00a0\u00d7\u00a0k)\n(e)\u00a0(3i)\u00a0\u00d7\u00a0(2j)\n(f)\u00a0(3i)\u00a0\u00d7\u00a0(2k)\n(g)\u00a0(i\u00a0\u00d7\u00a0x)\n(h)\u00a0(i\u00a0\u00d7\u00a0y)\n(i)\u00a0(v\u00a0\u00d7\u00a0j)\n(j)\u00a0(i\u00a0\u00d7\u00a0k)\n\n3. Dados os pontos A(2,1,-1),\u00a0B(3,0,1)\u00a0e\u00a0C(2,-1,-3),\u00a0 determine o ponto D tal que\nAB=BC\u00a0\u00d7\u00a0AC.\nBC=C-B\nB2=(2,1,-3)-(3,0,1)\nB3=(2,-1,4)\nAC=C-A\nAC=(2,-1,-3)-(2,1,-1)\nA=C-B\n 4. Determine o vetor \u007eZ tal que\u00a0\u007eZ=(1,4,-3)-2\u007eJ\u00a0\u2212\u007eX(4,-2,1)=(3,5,-2).\n\n5. Sejam os vetores \u007ew=(1,-2,1),\u007ev=(1,1,1)\u007ew=(1,0,-1).\n(a) Utilize o produto escalar para mostrar que os vetores s\u00e3o, dois a dois, ortogonais.\n(b) Utilize o produto vetorial para mostrar que o produto vetorial de quaisquer dois deles \u00e9 paralelo ao terceiro vetor.\n(c) Mostre que\u00a0\u007ew\u00a0\u00d7\u007ev=0\n\n6. Determine um vetor simultaneamente ortogonal aos vetores \u007ew+2\u007ev=\u007ew-\u007ew,\u00a0onde\u007ev=(-3,2,0)\n\n7. Dado \u007ew1=(2,1,1), determine os vetores \u007ew2 e \u007ew3 de modo que os tr\u00eas sejam mutuamente ortogonais.\n\n8. Dados os vetores \u007ew=(1,1,0)\u00a0e\u007ew=(1,-1,2), determine:\n(a) um vetor unit\u00e1rio simultaneamente ortogonal a \u007ew e \u007ew.\n(b) um vetor de m\u00f3dulo 5 ortogonal a simultaneamente ortogonal a \u007ew e \u007ew.\n\n9. Dados os vetores \u007ew=(3,-1,2)\u007ew=(2,-2,2,1), determine:\n(a) a \u00e1rea do paralelogramo determinado por \u007ew e \u007ew.\n(b) a altura do paralelogramo relativa \u00e0 base definida pelo vetor \u007ew.\n\n10. Calcule o vetor \u007ew=(1,-2,2) seja igual a\u00a0\u221a26.\n\n12. Sabendo que\u07b4|u|=6,\u00a0|v|=4\u00a0e 30\u00b0 o ângulo entre\u007ew e \u007ew, calcule:\n(a) a \u00e1rea do tri\u00e2ngulo determinado por \u007ew e \u007ew.\n(b) a \u00e1rea do paralelogramo determinado por \u007ew e (-\u007ew).\n\nRespostas\n1)\u00a0(a)\u00a0\u007ew\u00a0\u007ew=0\n(b)\u00a0(2\u007e)(3\u007ew)=0\n(c)\u007ew\u007ew\u00a0+\u007ewu\u007ew=0\n(d)\u007ew\u007ew\u00a0\u007ev\u007ew=0\n(e)\u007ew-\u007ev\u007ew\n\nj)\u007ewj\u007ew\n\n\n Lista 3\nProd. Vetorial.\n\n1)\n(a)\u007ew(\u007ew,\u007ew)=0\n(b)(2\u007ew)(2\u007ew)=0\n(c)(\u007ew\u007ew)(\u007ew)=0\nd)(\u007ew\u007ew)\u007ew(\u007ew\u007ew)=0\n(e)(\u007ew\u007ew)\u007ew\n\n3-3-2k=(2i+4j-k)\u007ew(\u007ei+\u007ek)\n(\u007ei-5j-k)(\u007ai+\u007ak)\n\n(-\u007ewi,0,-\u007ewk)\nF(\u007ew\u007ew\u007ew)\n(B1)\u007ew(B2)=3\u007ew=5\u007ew\n(-\u007ewk+15\u007ew2)\n(-2k-8k-33)\n\n1,-1,23,j\n\n {(u×v)×(u×w)}\nv_{uw}\n(i,5,k,j)\n(2,4,-1,2,4)\n[-1,0,1,1,0]\n\n(4i-3j)-(4k+2j)\n4i-3j, -(4k+6j+12j)\n(-10, -20, -5k)\n\n(i×(u×(v+w)))\n(2i+4j-k) + (-1-k)\n(3i+4s)\n\n(u×(v+w))\n(8j-1j+14)\n(21-14-j)\n(18,4,14) (i×v)·w\n(9,-1,4)\n(-1,0,1)\n(-9,-1,14)\n\n[(u×v)·w]\n(9,-1,4)\n(-1,0,1)\n(-9,-1,14)\n\n(u×(v×w))\n(4,-3,4)\n(3,-1,-2)\n\n(11,-3,-8)\n\n2)\nEfeito:\na)i×k\nij 0 0\n0 0 1\n0 1 0\n\nb)3×(2i)\nij 1 1 3\n0 0 0\n\nc)(3i)×(2k)\n0 0 0\n0 0 0\n\n1)\ni·(j×k)\n\n2)\n(3i)·(3)\n\n3) 3i×(-1)\n0\n\n1) j·(s×k)\n0 (s×k)×k\n0 0\n0 0\n\n(s×(5))×s\n4 4 5\n1 0 0\n0 1 0\n\n(5×s)×i\n0 0\n0 0 0\n\n2)(5×k)·i\n0 1 0\n0 0 1\n0 0 1\n\n(i·(s×s))\n0\n0\n0\n\n3)(i×(i×s))·k\n0 D. Arroba A(2,1,-1), B(3,0,1), C(2,-1,-3) Det. p. to D + to give AD = BC x AC\nBC = C-B\nBC = (2,-1,-3)-(3,0,1)\nBC = (-1,-1,-4)\nAC = C-A\nAC = (2,-1,-3)-(2,1,-1)\nPC = (0,-2,-2)\n\nD = (D-A) = (-6,-2,2) - (2,-1,-1)\nD = (-4,-1,1)\n\nX(1,4,-3) = -7\n= X(4,-2,-3) = (3,5,-2)\n(1 4 3)\n(1 3 4)\n(4 -2 3)\n(4i + (1 + 12)2k) - (-16k + 46i - 3)\n(-i - 11s + 14k) 5. Sejam u(1,-2,1), v(1,1,1) e w(1,0,-1)\n\na) Prova G. como prota a se um ling aunque\n{u.v(1,-2,1),(1,1,1)}\nul.w(1,2,-1),(1,0,-1)\n(u.w)(1,0,-1) = 0\n\n(u.v)(j,2,i)(1,0,4)\n\nb) Augmentas e paraleis as tensores.\nw(u.v)\nW = (1,2,1) • (1,3,9)\nW = (1,0,1)\n\nc) Mostre que u x v(y x w) = 0\n\n(i j k | i | 5)\n(1 | j | 1 | 1)\n\n(-1 | 0 | 1 | 0)\n(-i | 2 | 5 | k)\n\n4 - 2 = 0 6. Determine simit. ergagonal a u+w e u+v e v-w entre u = (-3,2,0) e v(0,-1,2)\n\n5. Prova v1 = (1,-2,1) def. v2 e v3 ergonais.\n\n8. Vamura u2 = (1,3,0) e v(1,-3,1,2) pertencem:\na) un. ortogant a u2 e v2\n\nb) sem. para o analog. s'ortog. u e v.\n m u d o 2 e n f. (x: (3,2,2) e U (0,1,1) Valores (3,-1,2) e (x1,-2,1) def. a) Usa Parale logramo para u e t. b) Usa Paraleg. Pal. base U. 3) Calcule valores para para p. para A3=(m,-3,j) V=(1,-2,2) = √26