Chapter_01.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6aa30998-f4e5-4d46-906b-57ebd516267d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Author: Brent Artuch\n",
    "# Date: 2024-11-08\n",
    "\n",
    "import sympy as sp\n",
    "import numpy as np\n",
    "from IPython.display import *\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "811f8c2b-327f-45f9-9d5c-b0e8188903ce",
   "metadata": {},
   "source": [
    "## Vector Analysis\n",
    "#### Vector Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67ad378d-63d5-4eaf-9efe-e89898b58b40",
   "metadata": {},
   "source": [
    "The properties of $\\mathbf{vector\\ addition}$ given two vectors $A$ and $B$ are:\n",
    "\\begin{align*}\n",
    "A+B&=B+A\\\\\n",
    "(A+B)+C&=A+(B+C)\\\\\n",
    "A-B&=A+(-B)\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a199479b-e396-4e34-8785-3bc5138dfa0b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 300x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Graph of vector addition\n",
    "A = np.array([1,2])\n",
    "B = np.array([2,1])\n",
    "C = A + B\n",
    "A_prime = np.array([2,1])\n",
    "B_prime = np.array([1,2])\n",
    "C_prime = A_prime + B_prime\n",
    "\n",
    "plt.figure(figsize=(3,3))\n",
    "plt.title(\"Vector Addition\")\n",
    "plt.annotate(r'$\\mathbf{B}$', xy=(1.5, 2.5))\n",
    "plt.annotate(r\"$\\mathbf{B}^'$\", xy=(2.5, 1.5))\n",
    "plt.annotate(r'$\\mathbf{A}$', xy=(0.5, 1.5))\n",
    "plt.annotate(r\"$\\mathbf{A}^'$\", xy=(1.5, 0.5))\n",
    "plt.annotate(r'$\\mathbf{C}$', xy=(1.3, 1.5))\n",
    "\n",
    "# Plot the vectors\n",
    "plt.arrow(0,0, A[0], A[1], head_width = 0.1, length_includes_head = True, width = 0.02, color='g') \n",
    "plt.arrow(A[0], A[1], B[0], B[1], head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b')\n",
    "plt.arrow(0,0, A_prime[0], A_prime[1], head_width = 0.1, length_includes_head = True, width = 0.02, color='g') \n",
    "plt.arrow(A_prime[0], A_prime[1], B_prime[0], B_prime[1], head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b')\n",
    "plt.arrow(0, 0, C[0], C[1], head_width = 0.1, length_includes_head = True, width = 0.02, color = 'r')\n",
    "\n",
    "plt.xlim(0,3.5)\n",
    "plt.ylim(0,3.5)\n",
    "\n",
    "plt.axis(\"off\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8587bad3-2355-4db0-8e3e-26b3dd5f02c0",
   "metadata": {},
   "source": [
    "The property of $\\mathbf{vector\\ scalar\\ multiplication}$ given two vectors $A$ and $B$ and a scalar $a$ are:\n",
    "\\begin{align*}\n",
    "a(A+B)&=aA+aB\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9a82eafe-168d-4603-8386-2c6a97f8b6d0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 300x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Graph of vector addition\n",
    "A = np.array([1,2])\n",
    "A_prime = 2*A\n",
    "\n",
    "plt.figure(figsize=(3,3))\n",
    "plt.title(\"Scalar Multiplication\")\n",
    "plt.annotate(r'$2\\mathbf{A}$', xy=(1.5, 2.5))\n",
    "plt.annotate(r\"$\\mathbf{A}$\", xy=(2.5, 1.5))\n",
    "\n",
    "# Plot the vectors\n",
    "plt.arrow(2,1, A[0], A[1], head_width = 0.1, length_includes_head = True, width = 0.02, color='g') \n",
    "plt.arrow(0, 0, A_prime[0], A_prime[1], head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b')\n",
    "\n",
    "plt.xlim(0,5)\n",
    "plt.ylim(0,5)\n",
    "\n",
    "plt.axis(\"off\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "634d68af-1af4-4c94-b635-56d3cbe6f367",
   "metadata": {},
   "source": [
    "The $\\mathbf{dot\\ product}$ of two vectors is:\n",
    "\\begin{align*}\n",
    "A\\cdot B\\equiv AB\\cos\\theta\n",
    "\\end{align*}\n",
    "where\n",
    "\\begin{align*}\n",
    "A\\cdot B &= B\\cdot A\\\\\n",
    "A\\cdot(B+C)&=A\\cdot B A\\cdot C\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c88c029c-d076-41b3-8503-bb26451a4930",
   "metadata": {},
   "source": [
    "Vectors are perpindicular (orthogonal) if:\n",
    "\\begin{align*}\n",
    "A\\cdot B=0\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ed1397e-1acd-48a3-bf45-77e3e39cd7d7",
   "metadata": {},
   "source": [
    "The $\\mathbf{cross\\ product}$ of two vectors is:\n",
    "\\begin{align*}\n",
    "A \\times B\\equiv AB\\sin\\theta\\hat{n}\n",
    "\\end{align*}\n",
    "where $\\hat{n}$ is a unit vector pointing perpendicular to the plane of $A$ and $B$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "632fd544-1857-4962-9887-60942ab2534c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Graph of cross product\n",
    "A = np.array([2,0,0])\n",
    "B = np.array([0,2,0])\n",
    "AcrossB = np.cross(A,B)\n",
    "\n",
    "fig = plt.figure(figsize=(4,4))\n",
    "ax = fig.add_subplot(projection=\"3d\")\n",
    "\n",
    "ax.text(1, 0, 0.25, r\"$\\mathbf{A}$\")\n",
    "ax.text(0, 1, 0.25, r\"$\\mathbf{B}$\")\n",
    "ax.text(0, 0.25, 2, r\"$\\mathbf{A}\\times\\mathbf{B}$\")\n",
    "\n",
    "ax.quiver(0,0,0, A[0], A[1], A[2], arrow_length_ratio=0.1, color='g') \n",
    "ax.quiver(0,0,0, B[0], B[1], B[2], arrow_length_ratio=0.1, color='y')\n",
    "ax.quiver(0,0,0, AcrossB[0], AcrossB[1], AcrossB[2], arrow_length_ratio=0.1, color='r')\n",
    "\n",
    "ax.set_xlim([0,3])\n",
    "ax.set_ylim([0,3])\n",
    "ax.set_zlim([0,4])\n",
    "\n",
    "# Rotate view\n",
    "ax.view_init(30, 50)\n",
    "ax.set_axis_off()\n",
    "ax.set_title(\"Cross Product\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98c123bb-eead-4a99-8515-99654176db21",
   "metadata": {},
   "source": [
    "Properties of the cross product:\n",
    "\\begin{align*}\n",
    "A\\times (B+C)&=(A\\times B)+(A\\times C)\\\\\n",
    "(B\\times A)&=-(A\\times B)\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a170b5e3-15f6-4b9c-83d6-dc1862e34172",
   "metadata": {},
   "source": [
    "If two vectors are parrallel, then:\n",
    "\\begin{align*}\n",
    "A\\times B =0\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "465313c4-a2c4-4c89-959b-fd4309e31e6b",
   "metadata": {},
   "source": [
    "## Component Form\n",
    "Let $\\hat{x}$, $\\hat{y}$, and $\\hat{z}$ be $\\mathbf{unit\\ vectors}$ parallel to the $x$, $y$, and $z$ axes respectively: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "dfe1f024-d958-4b61-a11e-d857cfe58b3c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Graph of unit vectors\n",
    "xhat = np.array([1,0,0])\n",
    "yhat = np.array([0,1,0])\n",
    "zhat = np.array([0,0,1])\n",
    "\n",
    "x = np.array([3,0,0])\n",
    "y = np.array([0,3,0])\n",
    "z = np.array([0,0,3])\n",
    "\n",
    "fig = plt.figure(figsize=(4,4))\n",
    "ax = fig.add_subplot(projection=\"3d\")\n",
    "\n",
    "ax.text(1, 0, 0.25, r\"$\\hat{x}$\")\n",
    "ax.text(0, 1, 0.25, r\"$\\hat{y}$\")\n",
    "ax.text(0, 0.25, 1, r\"$\\hat{z}$\")\n",
    "\n",
    "ax.quiver(0,0,0, x[0], x[1], x[2], arrow_length_ratio=0.1, color='b', linewidths=0.2) \n",
    "ax.quiver(0,0,0, y[0], y[1], y[2], arrow_length_ratio=0.1, color='b', linewidths=0.2)\n",
    "ax.quiver(0,0,0, z[0], z[1], z[2], arrow_length_ratio=0.1, color='b', linewidths=0.2)\n",
    "\n",
    "ax.quiver(0,0,0, xhat[0], xhat[1], xhat[2], arrow_length_ratio=0.1, color='g') \n",
    "ax.quiver(0,0,0, yhat[0], yhat[1], yhat[2], arrow_length_ratio=0.1, color='y')\n",
    "ax.quiver(0,0,0, zhat[0], zhat[1], zhat[2], arrow_length_ratio=0.1, color='r')\n",
    "\n",
    "ax.set_xlim([0,3])\n",
    "ax.set_ylim([0,3])\n",
    "ax.set_zlim([0,3])\n",
    "\n",
    "# Rotate view\n",
    "ax.view_init(30, 50)\n",
    "ax.set_axis_off()\n",
    "ax.set_title(\"Unit Vectors\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85a3b42b-e52e-451f-8166-c11a8fafd974",
   "metadata": {},
   "source": [
    "An arbitrary vector $A$ can be expanded in terms of these $\\mathbf{basis\\ vectors}$:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}=A_x\\hat{x}+A_y\\hat{y}+A_z\\hat{z}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ebac31bd-37b0-4b6c-a913-27c0d21ff6e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Graph of unit vectors\n",
    "xhat = np.array([1,0,0])\n",
    "yhat = np.array([0,1,0])\n",
    "zhat = np.array([0,0,1])\n",
    "\n",
    "x = np.array([3,0,0])\n",
    "y = np.array([0,3,0])\n",
    "z = np.array([0,0,3])\n",
    "\n",
    "A = np.array([2,2,2])\n",
    "\n",
    "Ax = A[0]\n",
    "Ay = A[1]\n",
    "Az = A[2]\n",
    "\n",
    "fig = plt.figure(figsize=(4,4))\n",
    "ax = fig.add_subplot(projection=\"3d\")\n",
    "\n",
    "ax.text(0, 0.25, 1, r\"$\\mathbf{A}$\")\n",
    "ax.text(1.5, 0, 0.5, r\"$A_x\\hat{x}$\")\n",
    "ax.text(2, 1, 0.25, r\"$A_y\\hat{y}$\")\n",
    "ax.text(0, 1.5, 0.25, r\"$A_Z\\hat{z}$\")\n",
    "\n",
    "ax.quiver(0,0,0, x[0], x[1], x[2], arrow_length_ratio=0.1, color='b', linewidths=0.2) \n",
    "ax.quiver(0,0,0, y[0], y[1], y[2], arrow_length_ratio=0.1, color='b', linewidths=0.2)\n",
    "ax.quiver(0,0,0, z[0], z[1], z[2], arrow_length_ratio=0.1, color='b', linewidths=0.2)\n",
    "\n",
    "ax.quiver(0,0,0, A[0], A[1], A[2], arrow_length_ratio=0.1, color='purple')\n",
    "\n",
    "ax.quiver(0,0,0, Ax*xhat[0], Ax*xhat[1], Ax*xhat[2], arrow_length_ratio=0.1, color='g') \n",
    "ax.quiver(Ax*xhat[0], Ax*xhat[1], Ax*xhat[2], Ay*yhat[0], Ay*yhat[1], Ay*yhat[2], arrow_length_ratio=0.1, color='y')\n",
    "ax.quiver(\n",
    "    Ax*xhat[0]+Ay*yhat[0], Ax*xhat[1]+Ay*yhat[1], Ax*xhat[2]+Ay*yhat[2], Az*zhat[0], Az*zhat[1], Az*zhat[2], \n",
    "    arrow_length_ratio=0.1, color='r')\n",
    "\n",
    "ax.set_xlim([0,3])\n",
    "ax.set_ylim([0,3])\n",
    "ax.set_zlim([0,3])\n",
    "\n",
    "# Rotate view\n",
    "ax.view_init(10,20)\n",
    "ax.set_axis_off()\n",
    "ax.set_title(\"Basis Vectors\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02907900-15ba-4b0d-b44b-d1d36b7348ca",
   "metadata": {},
   "source": [
    "Given two vectors $\\mathbf{A}$ and $\\mathbf{B}$, we can express the addition of these two vectors as:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}+\\mathbf{B}&=(A_x\\hat{x}+A_y\\hat{y}+A_z\\hat{z})+(B_x\\hat{x}+B_y\\hat{y}+B_z\\hat{z})\\\\\n",
    "&=(A_x+B_x)\\hat{x}+(A_y+B_y)\\hat{y}+(A_z+B_z)\\hat{z}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7313b832-86e2-4b1f-a80b-97bc66633f5d",
   "metadata": {},
   "source": [
    "If we have a scalar $a$, we multiply each component:\n",
    "\\begin{align*}\n",
    "a\\mathbf{A}=(aA_x)\\hat{x}+(aA_y)\\hat{y}+(aA_z)\\hat{z}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14cf5670-c4b1-400d-9311-63d926ad2e57",
   "metadata": {},
   "source": [
    "The dot product of these two vectors can be expressed as:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}\\cdot\\mathbf{B}&=(A_x\\hat{x}+A_y\\hat{y}+A_z\\hat{z})\\cdot(B_x\\hat{x}+B_y\\hat{y}+B_z\\hat{z})\\\\\n",
    "&=A_xB_x+A_yB_y+A_zB_z\n",
    "\\end{align*}\n",
    "because the dot product of any unit vector with itself is $1$ and the dot product of any unit vector with another unit vector in the same basis is $0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2dd7eddf-eada-4bfe-97b1-17360302aa32",
   "metadata": {},
   "source": [
    "We can use the dot product of a vector with itself to find its magnitude. Thus, we have:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}\\cdot\\mathbf{A}&=A_xA_x+A_yA_y+A_zA_z\\\\\n",
    "\\mathbf{A}^2&=A_x^2+A_y^2+A_z^2\\\\\n",
    "\\mathbf{A}&=\\sqrt{A_x^2+A_y^2+A_z^2}\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb227ddc-cd43-4521-a4f8-ba700888a1c2",
   "metadata": {},
   "source": [
    "The cross product of these two vectors can be expressed as:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}\\times\\mathbf{B}&=(A_x\\hat{x}+A_y\\hat{y}+A_z\\hat{z})\\times(B_x\\hat{x}+B_y\\hat{y}+B_z\\hat{z})\\\\\n",
    "&=(A_yB_z-A_zB_y)\\hat{x}+(A_zB_x-A_xB_z)\\hat{y}+(A_xB_y-A_yB_x)\\hat{z}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78502da9-a106-4844-a9f8-f20466e5f896",
   "metadata": {},
   "source": [
    "This can be simplified as a determinant:\n",
    "\\begin{align*}\n",
    "A\\times B=\\left|\n",
    "\\begin{matrix}\n",
    "\\hat{x} & \\hat{y} & \\hat{z}\\\\\n",
    "A_x & A_y & A_z\\\\\n",
    "B_x & B_y & B_z\\\\\n",
    "\\end{matrix}\\right|\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a544280-fd35-4eb1-bdb3-1e444fc2a5c5",
   "metadata": {},
   "source": [
    "Example 1.2 - Find the angle between the face diagnols of a cube.<br>\n",
    "Answer - If we use a cube that is length $1$ on all sides, the face diagnols $\\mathbf{A}$ and $\\mathbf{B}$ will be:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}=1\\hat{x}+0\\hat{y}+1\\hat{Z}\\\\\n",
    "\\mathbf{B}=0\\hat{x}+1\\hat{y}+1\\hat{Z}\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "964446f8-b6df-4cde-a9cb-a7770217b513",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create a figure and an axes object.\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "x = np.array([3,0,0])\n",
    "y = np.array([0,3,0])\n",
    "z = np.array([0,0,3])\n",
    "\n",
    "xhat = np.array([1,0,0])\n",
    "yhat = np.array([0,1,0])\n",
    "zhat = np.array([0,0,1])\n",
    "\n",
    "A = xhat + zhat\n",
    "B = yhat + zhat\n",
    "\n",
    "# Define the vertices of the cube.\n",
    "vertices = np.array([\n",
    "    [0, 0, 0],\n",
    "    [1, 0, 0],\n",
    "    [1, 1, 0],\n",
    "    [0, 1, 0],\n",
    "    [0, 0, 1],\n",
    "    [1, 0, 1],\n",
    "    [1, 1, 1],\n",
    "    [0, 1, 1],\n",
    "])\n",
    "\n",
    "# Define the edges of the cube.\n",
    "edges = [\n",
    "    [0, 1], [1, 2], [2, 3], [3, 0],\n",
    "    [4, 5], [5, 6], [6, 7], [7, 4],\n",
    "    [0, 4], [1, 5], [2, 6], [3, 7],\n",
    "]\n",
    "\n",
    "# Plot the edges of the cube.\n",
    "for edge in edges:\n",
    "    ax.plot(vertices[edge, 0], vertices[edge, 1], vertices[edge, 2], '--', color='black')\n",
    "\n",
    "ax.quiver(0,0,0, x[0], x[1], x[2], arrow_length_ratio=0.1, color='b', linewidths=0.2) \n",
    "ax.quiver(0,0,0, y[0], y[1], y[2], arrow_length_ratio=0.1, color='b', linewidths=0.2)\n",
    "ax.quiver(0,0,0, z[0], z[1], z[2], arrow_length_ratio=0.1, color='b', linewidths=0.2)\n",
    "\n",
    "ax.quiver(0,0,0, A[0], A[1], A[2], arrow_length_ratio=0.1, color='g') \n",
    "ax.quiver(0,0,0, B[0], B[1], B[2], arrow_length_ratio=0.1, color='y')\n",
    "\n",
    "ax.text(0.1, 0, 0.2, r\"$\\theta$\", fontsize=20)\n",
    "ax.text(0.7, 0, 0.4, r\"$\\mathbf{A}$\", fontsize=16)\n",
    "ax.text(0, 0.7, 0.5, r\"$\\mathbf{B}$\", fontsize=16)\n",
    "\n",
    "# Rotate view\n",
    "ax.view_init(20,20)\n",
    "ax.set_axis_off()\n",
    "ax.set_title(\"Angle Between Face Diagonals\")\n",
    "\n",
    "\n",
    "# Show the plot.\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f6b217b3-6b9a-4618-bd70-101c7f445774",
   "metadata": {},
   "source": [
    "Since we need to find the angle $\\theta$, we need to use the geometric interpretation of the dot product to solve for $\\theta$. So, the first thing we need to do is find the magnitudes of $\\mathbf{A}$ and $\\mathbf{B}$:\n",
    "\\begin{align*}\n",
    "\\mathbf{A}=\\sqrt{1^2+0^2+1^2}=\\sqrt{2}\\\\\n",
    "\\mathbf{B}=\\sqrt{0^2+1^2+1^2}=\\sqrt{2}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2430a741-7c8f-4a12-a7c1-9b9442c20240",
   "metadata": {},
   "source": [
    "Recall that the geometric interpretation of the dot product is:\n",
    "\\begin{align*}\n",
    "\\mathbf{AB}\\cos\\theta\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dadf553b-3ad0-404e-b50c-f2284edb192d",
   "metadata": {},
   "source": [
    "So, after pluging in the values we get:\n",
    "\\begin{align*}\n",
    "\\sqrt{2}\\sqrt{2}\\cos\\theta=2\\cos\\theta\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ce51640-7651-4593-b353-37f28e393b68",
   "metadata": {},
   "source": [
    "Therefore, \n",
    "\\begin{align*}\n",
    "\\cos\\theta=\\frac{1}{2}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39f7e4b3-a6fe-4c28-921b-25d1d251b211",
   "metadata": {},
   "source": [
    "and if we reference the unit circle:\n",
    "\\begin{align*}\n",
    "\\theta=\\frac{\\pi}{3}=60\\degree\n",
    "\\end{align*}"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "electrodynamics_venv",
   "language": "python",
   "name": "electrodynamics_venv"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}