angEntreVetores.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "f4b84061-e180-42b3-b787-4e7e489ab63a",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ba1bf2cc-d034-4a3a-a9b5-9a2db222013a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plotVectors(vecs, cols, alpha=1):\n",
    "    \n",
    "    plt.figure()\n",
    "    plt.axvline(x=0, color='#A9A9A9', zorder=0)\n",
    "    plt.axhline(y=0, color='#A9A9A9', zorder=0)\n",
    "\n",
    "    for i in range(len(vecs)):\n",
    "        X = np.concatenate([[0, 0], vecs[i]])\n",
    "        plt.quiver([X[0]],\n",
    "                   [X[1]],\n",
    "                   [X[2]],\n",
    "                   [X[3]],\n",
    "                   angles='xy', scale_units='xy', scale=1, color=cols[i],\n",
    "                  alpha=alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "220e8354-5e9d-446e-8e37-d90a2f218b77",
   "metadata": {},
   "outputs": [],
   "source": [
    "def ang_2vet(v, u):\n",
    "    v_interno_u = v.dot(u)\n",
    "    \n",
    "    vn = np.linalg.norm(v)\n",
    "    un = np.linalg.norm(u)\n",
    "    \n",
    "    r = v_interno_u/(vn*un)\n",
    "    \n",
    "    ang = np.arccos(r) #em radianos\n",
    "    \n",
    "    return 180/np.pi*ang #em graus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "9b6900ed-4030-4ae7-bfca-15896224af68",
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.array([0, 1])\n",
    "v = np.array([1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "99c1d8ad-e8bd-4337-ab5f-ea64bc0425ce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.0, 5.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "red = 'red'\n",
    "blue = 'blue'\n",
    "\n",
    "plotVectors([u, v], [red, blue])\n",
    "plt.xlim(-1, 5)\n",
    "plt.ylim(-1, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "01f175ec-383a-45af-bb76-28bef09e6a3c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ang_2vet(u, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "74270912-5b42-4059-8704-225a4927a397",
   "metadata": {},
   "outputs": [],
   "source": [
    "u2 = np.array([1, 1])\n",
    "u3 = np.array([1, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "39b7c249-2f06-46e2-89f0-65805acac17b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.0, 2.0)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYQAAAD8CAYAAAB3u9PLAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuNCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8QVMy6AAAACXBIWXMAAAsTAAALEwEAmpwYAAAV30lEQVR4nO3de5RdZX3/8feXhIskKQiEEAIB1BRJuPwShwCVRQlChRQLdBUWuqq0oDFKltAbRmn52dYb6rIWuYSgVKgUYVWRAAmYYACxwC8JBSQENAKWmBQCaAjILeT7+2PvOOMwk5nk7Dn7nJn3a62z5uy9nznPd7OZfObZl2ciM5EkaZu6C5AktQYDQZIEGAiSpJKBIEkCDARJUslAkCQBFQRCROwdEYsjYkVELI+Ic3poExFxUUSsjIiHImJKo/1Kkqo1vILP2AD8TWbeHxGjgGURsTAzH+nS5gRgQvk6DLis/CpJahENjxAyc01m3l++Xw+sAMZ1a3YScHUW7gV2joixjfYtSapOFSOE34qIfYHJwH3dNo0DnuqyvKpct6aHz5gBzAAYMWLEu975zndWWaKaYP369QCMGjWq5kqkoWfZsmXPZuborfneygIhIkYC3wXOzcwXum/u4Vt6nDMjM+cCcwE6Ojpy6dKlVZWoJlm8eDEA06ZNq7kSaeiJiF9s7fdWcpdRRGxLEQbXZOb3emiyCti7y/JewOoq+pYkVaOKu4wC+CawIjO/2kuzecCHyruNDgfWZeabThdJkupTxSmjdwMfBH4SEQ+U6z4NjAfIzDnAfGA6sBL4DfCXFfQrSapQw4GQmXfT8zWCrm0SOLvRviRJA8cnlSVJgIEgSSoZCJIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSSUDQZIEGAiSpJKBIEkCDARJUslAkCQBBoIkqWQgSJIAA0GSVDIQJEmAgSBJKhkIkiTAQJAklQwESRJgIEiSSgaCJAkwECRJJQNBkgQYCJKkkoEgSQIqCoSIuDIinomIh3vZfnRErIuIB8rXBVX0K0mqzvCKPudbwMXA1Ztp86PMPLGi/iRJFatkhJCZdwHPV/FZkqR6NPMawhER8WBELIiISU3sV5LUD1WdMurL/cA+mfliREwHvg9M6KlhRMwAZgCMHz++SeVJkpoyQsjMFzLzxfL9fGDbiNitl7ZzM7MjMztGjx7djPIkSTQpECJij4iI8v3Ust/nmtG3JKl/KjllFBHXAkcDu0XEKuD/AtsCZOYc4M+Aj0XEBuBl4PTMzCr6liRVo5JAyMz397H9YorbUiVJLconlSVJgIEgSSoZCJIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSSUDQZIEGAiSpJKBIEkCDARJUslAkCQBBoIkqWQgSJIAA0GSVDIQJEmAgSBJKhkIkiTAQJAklQwESRJgIEiSSgaCJAkwECRJJQNBkgQYCJKkkoEgSQIMBElSqZJAiIgrI+KZiHi4l+0RERdFxMqIeCgiplTRrySpOlWNEL4FHL+Z7ScAE8rXDOCyivqVJFWkkkDIzLuA5zfT5CTg6izcC+wcEWOr6FtSxV5+ue4KVJNmXUMYBzzVZXlVue5NImJGRCyNiKVr165tSnHSkLZxIyxZAp/5DBx5JCxbVndFqsnwJvUTPazLnhpm5lxgLkBHR0ePbSQ1aN06WLgQbrkFFiyAp58u1l91VREKGpKaFQirgL27LO8FrG5S35Iy4dFHiwC45Ra4+27YsOF325x/PnzoQ/XUp5bQrECYB8yKiO8AhwHrMnNNk/qWhrb582HWLHjiid7bnHoq/NM/Na8mtaSqbju9FrgH2D8iVkXEWRExMyJmlk3mA48DK4ErgI9X0a+kfpg+HS6+GN7xjp63T51anCraxseShrpKRgiZ+f4+tidwdhV9SdpCmbB+fXHdoLvx4+HGG+Etb2l+XWo5zTplJKkOq1fDxz4G8+a9edvIkXDTTbDHHs2vSy3JMaI0GGXCN74BEyd2hsEuu8DVV8MOOxSnh667Dg4+uN461VIMBGmwefxxOPZY+MhHOk8TnXYarFgBH/wg7LUX/Mu/FNcWpC48ZSQNFm+8ARddVNw+uulp47Fj4dJL4eSTO9t97nPFXUVSNwaCNBg8/DB8+MNw332d6z78Yfjyl2HnnX+37WmnNbU0tQ9PGUnt7LXX4B//EaZM6QyD/faDRYvgiiveHAbSZjhCkNrVkiVw5pnF6AAgAs49F/75n2HEiFpLU3syEKR285vfwAUXFBeGN24s1k2cCN/8Jhx+eL21qa0ZCFI7ueOO4trAz39eLA8fDp/+dPHafvtaS1P7MxCkdrBuHZx3Hsyd27muowOuvBIOOqi+ujSoeFFZanU33QSTJnWGwQ47wFe+AvfcYxioUo4QpFa1di2ccw5ce23nuj/8w+IJ5N4mqpMa4AhBajWZ8B//AQcc0BkGo0bB5ZfDD39oGGjAOEKQWsmqVcVkdDff3Lnuj/8Y5swpppyQBpAjBKkVbNxYjAAmTuwMg912K0YKN91kGKgpHCFIdVu5spiI7o47Otd94APwta/B6NF1VaUhyBGCVJcNG4q7hQ46qDMMxo0rpqu+5hrDQE3nCEGqw0MPwVlnwdKlnes++lG48ELYaaf66tKQZiBIzfTqq/D5zxevDRuKdW9/ezER3bRp9damIc9AkJrl3nuLUcEjjxTL22wDf/3XxWylO+5Yb20SBoI08F56Cf7hH4qLxJnFuoMOKiajO/TQWkuTujIQpIF0++3FHURPPFEsb7st/P3fw+zZsN129dYmdWMgSAPh17+Gv/u7YpqJTQ47rBgVTJpUW1nS5njbqVS1G28sHjDbFAZveQt89avw4x8bBmppjhCkqjz9NHziE3D99Z3rjjmmuIPobW+rry6pnxwhSI3KhG9/uxgVbAqDnXYqRgiLFhkGahuOEKRG/M//wMyZsGBB57qTToJLL4U996yvLmkrVDJCiIjjI+KxiFgZEbN72H50RKyLiAfK1wVV9CvVZuPG4h/9SZM6w2D0aLjuOrjhBsNAbanhEUJEDAMuAY4DVgFLImJeZj7SremPMvPERvuTavfTnxZ/1/hHP+pc9+d/XjxnsOuutZUlNaqKEcJUYGVmPp6ZrwHfAU6q4HOl1rJhQzHX0MEHd4bBXnvBLbfAv/+7YaC2V0UgjAOe6rK8qlzX3RER8WBELIiIXu+9i4gZEbE0IpauXbu2gvKkCjz4YPEcwezZxXxEAB//OCxfDtOn11ubVJEqAiF6WJfdlu8H9snMQ4CvA9/v7cMyc25mdmRmx2in/1XdXnmleLK4owPuv79YN2EC3HknXHIJ/N7v1VufVKEqAmEVsHeX5b2A1V0bZOYLmfli+X4+sG1E7FZB39LA+a//gsmT4XOfK04XDRsGn/xkMVo46qi6q5MqV0UgLAEmRMR+EbEdcDowr2uDiNgjIqJ8P7Xs97kK+paq9+KLcM45cOSR8OijxbpDDoH77oMvfrF48lgahBq+yygzN0TELOA2YBhwZWYuj4iZ5fY5wJ8BH4uIDcDLwOmZ2f20klS/H/wAZsyAX/yiWN5uO7jgAjjvvGJiOmkQq+TBtPI00Pxu6+Z0eX8xcHEVfUkD4le/Kv42wbe+1bnuiCOKyegOOKC2sqRmcuoK6XvfK6ad2BQGO+4I//qvxa2lhoGGEKeu0ND1v/8Ls2bBd7/bue6442DuXNh339rKkuriCEFDTyZcdVUxKtgUBjvvDP/2b3DbbYaBhixHCBpannwSPvrR4uLxJqecUjxTMHZsbWVJrcARgoaGjRvh61+HAw/sDIMxY+A//7O4hmAYSI4QNAQ8+mgxGd2Pf9y57owzir9itssu9dUltRhHCBq8Xn8dPv/54qGyTWEwfjzcemtxR5FhIP0ORwganP77v+HMM+GBB4rlCDj77CIgRo2qtTSpVTlC0ODyyivwqU/BoYd2hsH++8NddxXXEAwDqVeOEDR43H03nHVW8QdsoJiM7rzziqkndtih3tqkNmAgqP2tX1+MCi65pHPd5MnFtBOTJ9dXl9RmPGWk9nbrrcWtpJvCYPvt4QtfKGYmNQykLeIIQe3pueeKyeiuvrpz3ZFHwje+UVwzkLTFHCGovWQWD5NNnNgZBiNHwsUXF3/FzDCQtpojBLWPNWuKW0dvuKFz3XvfC5dfDvvsU19d0iDhCEGtLxOuvLIYFWwKg7e+tZigbsECw0CqiCMEtbYnnij+gtmiRZ3rTj21eKZgzJj66pIGIUcIak1vvFH8kZoDD+wMgz32KCaiu/56w0AaAI4Q1HoeeaSYjO6eezrXnXkmfOUrxakiSQPCEYJax+uvw2c/Wzw/sCkM9t0XFi4sHjIzDKQB5QhBrWHZsmIU8NBDxXIEfOITRUCMHFlvbdIQ4QhB9Xr5ZfjkJ2Hq1M4wOOCAYrrqr33NMJCayBGC6nPnncW1gpUri+Xhw4s5ic4/v5iCQlJTGQhqvhdeKEYFc+Z0rnvXu4rrBIccUl9d0hDnKSM11/z5MGlSZxjssAN86Utw772GgVQzRwhqjmefhXPPhWuu6Vx31FFwxRXw+79fW1mSOjlC0MDKhOuuK6ad2BQGo0bBZZfB4sWGQRMsWVJMDiv1pZJAiIjjI+KxiFgZEbN72B4RcVG5/aGImFJFv2pxv/wlnHwynH46rF1brJs+HZYvh5kzYRt/H2mG1ath992L2cEvvLB47i+z7qrUihr+iYyIYcAlwAnAROD9ETGxW7MTgAnlawZwWaP9qoVlMvbmm4tRwbx5xbpdd4Vvfxtuvhn23rve+oaYP/kTOPjg4k7e2bOLSzjveEdxBm/RInjttborVKuIbPBXhYg4AvhMZr63XP4UQGZ+oUuby4E7MvPacvkx4OjMXLO5z95///1zTtc7UdQWtvvhDznvs8fzCMXvBTl8OG9sv33xsJlq8eqrw3j99Z5//xsxYgMdHc9zxBHPcthhz7PTTq83uTpV6ZhjjlmWmR1b871VXFQeBzzVZXkVcFg/2owD3hQIETGDYhTBGCcwa0vPTJ7M8zuM4devlFNNbChfakkvvTScO+/cnXvu2ZVDD32ev/iLJ3j721+quyzVoIpA6OnXvu7Djv60KVZmzgXmAnR0dOS0adMaq05Nt3jxYo465RWOHfmqD5i1iIUL4bHH3rx+7Fg48UR43/vgPe8Zxo47jgZGN70+tYYqAmEV0PWk8F7A6q1oo0Hk9I+sZdq0A+suQ8C6dcXlm02mTCkC4H3vK+YR9Nq+NqkiEJYAEyJiP+CXwOnAB7q1mQfMiojvUJxOWtfX9QNJ1bjiiuIOoxNPLF7jxtVdkVpVw4GQmRsiYhZwGzAMuDIzl0fEzHL7HGA+MB1YCfwG+MtG+5XUP3/1V/C3f1t3FWoHlTypnJnzKf7R77puTpf3CZxdRV+StsywYXVXoHbh2UNJEmAgSJJKBoIkCTAQJEklA0GSBBgIkqSSgSBJAgwESVLJQJAkAQaCJKlkIEiSAANBklQyECRJgIEgSSoZCJIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSSUDQZIEGAiSpJKBIEkCDARJUslAkCQBBoIkqWQgSJIAGN7IN0fELsB1wL7Ak8BpmfmrHto9CawH3gA2ZGZHI/1KkqrX6AhhNnB7Zk4Abi+XezMtM/+PYSBJranRQDgJuKp8fxVwcoOfJ0mqSaOBMCYz1wCUX3fvpV0CP4iIZRExo8E+JUkDoM9rCBGxCNijh03nb0E/787M1RGxO7AwIh7NzLt66W8GMANg/PjxW9CFJKkRfQZCZh7b27aIeDoixmbmmogYCzzTy2esLr8+ExE3AFOBHgMhM+cCcwE6Ojqy712QJFWh0VNG84AzyvdnADd2bxARIyJi1Kb3wB8BDzfYrySpYo0GwheB4yLiZ8Bx5TIRsWdEzC/bjAHujogHgf8H3JKZtzbYrySpYg09h5CZzwHv6WH9amB6+f5x4JBG+pEkDTyfVJYkAQaCJKlkIEiSAANBklQyECRJgIEgSSoZCJIkwECQJJUMBEkSYCBIkkoGgiQJMBAkSSUDQZIEGAiSpJKBIEkCDARJUslAkCQBBoIkqWQgSJIAA0GSVDIQJEmAgSBJKhkIkiTAQJAklQwESRJgIEiSSgaCJAkwECRJpYYCISJOjYjlEbExIjo20+74iHgsIlZGxOxG+pQkDYxGRwgPA38K3NVbg4gYBlwCnABMBN4fERMb7FeSVLHhjXxzZq4AiIjNNZsKrMzMx8u23wFOAh5ppG9JUrUaCoR+Ggc81WV5FXBYb40jYgYwo1x8NSIeHsDa6rQb8GzdRQwg96+9uX/ta/+t/cY+AyEiFgF79LDp/My8sR999DR8yN4aZ+ZcYG7Z99LM7PXaRDsbzPsG7l+7c//aV0Qs3drv7TMQMvPYrf3w0ipg7y7LewGrG/xMSVLFmnHb6RJgQkTsFxHbAacD85rQryRpCzR62+kpEbEKOAK4JSJuK9fvGRHzATJzAzALuA1YAVyfmcv72cXcRuprcYN538D9a3fuX/va6n2LzF5P50uShhCfVJYkAQaCJKnUMoEw2KfBiIhdImJhRPys/PrWXto9GRE/iYgHGrl9rFn6Oh5RuKjc/lBETKmjzq3Vj/07OiLWlcfrgYi4oI46t0ZEXBkRz/T2rM8gOHZ97V87H7u9I2JxRKwo/908p4c2W378MrMlXsABFA9U3AF09NJmGPBz4G3AdsCDwMS6a+/n/n0JmF2+nw1c2Eu7J4Hd6q63n/vU5/EApgMLKJ5HORy4r+66K96/o4Gb6651K/fvKGAK8HAv29v22PVz/9r52I0FppTvRwE/reJnr2VGCJm5IjMf66PZb6fByMzXgE3TYLSDk4CryvdXASfXV0pl+nM8TgKuzsK9wM4RMbbZhW6ldv7/rU+ZeRfw/GaatPOx68/+ta3MXJOZ95fv11PcwTmuW7MtPn4tEwj91NM0GN3/I7SqMZm5BoqDCezeS7sEfhARy8ppPFpZf45HOx+z/tZ+REQ8GBELImJSc0prinY+dv3V9scuIvYFJgP3ddu0xcevGXMZ/Vazp8Fots3t3xZ8zLszc3VE7A4sjIhHy990WlF/jkdLH7M+9Kf2+4F9MvPFiJgOfB+YMNCFNUk7H7v+aPtjFxEjge8C52bmC9039/Atmz1+TQ2EHOTTYGxu/yLi6YgYm5lrymHbM718xury6zMRcQPFaYtWDYT+HI+WPmZ96LP2rj+EmTk/Ii6NiN0yczBMnNbOx65P7X7sImJbijC4JjO/10OTLT5+7XbKqJ2nwZgHnFG+PwN404goIkZExKhN74E/ovibE62qP8djHvCh8o6Hw4F1m06dtYE+9y8i9ogo5n+PiKkUP1PPNb3SgdHOx65P7Xzsyrq/CazIzK/20mzLj1/dV8u7XBE/hSLRXgWeBm4r1+8JzO925fynFHd/nF933Vuwf7sCtwM/K7/u0n3/KO5mebB8LW+H/evpeAAzgZnl+6D4A0k/B35CL3eQteqrH/s3qzxWDwL3An9Qd81bsG/XAmuA18ufvbMG2bHra//a+dgdSXH65yHggfI1vdHj59QVkiSg/U4ZSZIGiIEgSQIMBElSyUCQJAEGgiSpZCBIkgADQZJU+v/i24zcVXjEqAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "red = 'red'\n",
    "blue = 'blue'\n",
    "\n",
    "plotVectors([u2, u3], [red, blue])\n",
    "plt.xlim(-1, 2)\n",
    "plt.ylim(-1, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "15781591-ba11-4a92-a616-9718e57015f3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "45.00000000000001"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ang_2vet(u2, u3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "afa3d8b4-5ebd-4344-898f-e30615aba43e",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}