Vector and Scalar Multiplication.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f588e125",
   "metadata": {},
   "source": [
    "# Vector and Scalar Multiplication "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "44bb8f07",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Modules\n",
    "\n",
    "import matplotlib.pyplot as pl\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "610b1bc5",
   "metadata": {},
   "source": [
    "## Vector *  Scalar means it shows the elements of the vector scalar number of time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "520ff4b5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 4, 5, 6, 3, 4, 5, 6]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vec = [3,4,5,6]\n",
    "s=2\n",
    "\n",
    "vec*s"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f5e1d83",
   "metadata": {},
   "source": [
    "## vec * s while vec=np.array(vec) then it miltiplies the element of the vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9f722cf7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6,  8, 10, 12])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vec = np.array([3,4,5,6])\n",
    "s=2\n",
    "\n",
    "vec*s"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99b0b690",
   "metadata": {},
   "source": [
    "## ploting vectors in graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1c409a02",
   "metadata": {},
   "outputs": [],
   "source": [
    "vec2d= np.array([1,2])\n",
    "s1=2\n",
    "s2=5\n",
    "s3=-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "96614ccf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pl.plot([0,vec2d[0]],[0,vec2d[1]],'bs-', label='v' )\n",
    "pl.plot([0,s1*vec2d[0]],[0,s1*vec2d[1]],'ro-',label='v*s1' )\n",
    "pl.plot([0,s2*vec2d[0]],[0,s2*vec2d[1]],'kp-',label='v*s2' )\n",
    "pl.plot([0,s3*vec2d[0]],[0,s3*vec2d[1]],'g*-',label='v*s3' )\n",
    "\n",
    "\n",
    "# pl.axis('square')\n",
    "pl.xlim([-4,4])\n",
    "pl.ylim([-4,4])\n",
    "pl.grid()\n",
    "pl.legend()\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eef6eb87",
   "metadata": {},
   "source": [
    "## creating 2 vectors\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8a1506b3",
   "metadata": {},
   "outputs": [],
   "source": [
    "v1= np.array([4,7,8,5])\n",
    "v2= np.array([7,1,2,3])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02339a86",
   "metadata": {},
   "source": [
    "## plotting each vector\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7b0c44fc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pl.plot([0,v1[0]],[0,v1[1]],[0,v1[2]],[0,v1[3]],'ro-',label='v1')\n",
    "pl.plot([0,v2[0]],[0,v2[1]],[0,v2[2]],[0,v2[3]],'bo-',label='v2')\n",
    "\n",
    "pl.grid()\n",
    "pl.legend()\n",
    "pl.xlim(-10,10)\n",
    "pl.ylim(-10,10)\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3fea6822",
   "metadata": {},
   "source": [
    "## V1 + V2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "578ad2fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "v3=v1+v2\n",
    "\n",
    "pl.plot([0,v3[0]],[0,v3[1]],[0,v3[2]],[0,v3[3]],'ro-',label='v1')\n",
    "\n",
    "pl.grid()\n",
    "pl.legend()\n",
    "pl.xlim(-10,10)\n",
    "pl.ylim(-10,10)\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d39a79a4",
   "metadata": {},
   "source": [
    "## v1-v2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "23893c36",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "v4=v1-v2\n",
    "\n",
    "pl.plot([0,v4[0]],[0,v4[1]],[0,v4[2]],[0,v4[3]],'ro-',label='v1')\n",
    "\n",
    "pl.grid()\n",
    "pl.legend()\n",
    "pl.xlim(-10,10)\n",
    "pl.ylim(-10,10)\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5281a9d",
   "metadata": {},
   "source": [
    "## v1 * 4 +v2/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3df9fdf4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "v5=(v1*4)+(v2/2)\n",
    "\n",
    "pl.plot([0,v5[0]],[0,v5[1]],[0,v5[2]],[0,v5[3]],'ro-',label='v1')\n",
    "\n",
    "pl.grid()\n",
    "pl.legend()\n",
    "pl.xlim(0,50)\n",
    "pl.ylim(0,50)\n",
    "pl.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a9622047",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.10.5 64-bit",
   "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": "3.10.5"
  },
  "vscode": {
   "interpreter": {
    "hash": "2cd637df22d7153cd6f5a2f9dca8aa8968afa145fb63c8e9d35b2f401c1fb46c"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}