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    "# A simple 3D vector class\n",
    "\n",
    "A 3D vector has 3 coordinates along the X, Y and Z-axis and is commonly used to represent geometry in computer graphics. In it simple form here it supports only one operation `add` that computes the component-wise sum of two vectors. \n",
    "\n",
    "$$\n",
    "\\vec{a} + \\vec{b} \n",
    "= \n",
    "\\begin{pmatrix} \n",
    "a_x \\\\ a_y \\\\ a_z\n",
    "\\end{pmatrix}\n",
    "+\n",
    "\\begin{pmatrix} \n",
    "b_x \\\\ b_y \\\\ b_z\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix} \n",
    "a_x + b_x \\\\ a_y + b_y \\\\ a_z + b_z\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "\n",
    "Implementations like below can often be found in the wild:"
   ]
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  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "public class Vec3 {\n",
    "    private double x, y, z;\n",
    "\n",
    "    public Vec3(double x, double y, double z) {\n",
    "        this.x = x;\n",
    "        this.y = y;\n",
    "        this.z = z;\n",
    "    }\n",
    "\n",
    "    public Vec3 add(Vec3 v) {\n",
    "        x += v.x;\n",
    "        y += v.y;\n",
    "        z += v.z;\n",
    "        return this;\n",
    "    }\n",
    "\n",
    "    @Override\n",
    "    public String toString() {\n",
    "        return \"(\" + x + \", \" + y + \", \" + z + \")\";\n",
    "    }\n",
    "}\n",
    "\n",
    "System.out.println(new Vec3(1.0, 2.0 ,3.0));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This implementation has a [*fluent interface*](https://en.wikipedia.org/wiki/Fluent_interface) for it's `add` method that allows easy chaining of operations on a vector. For example the addition of 3 vectors can be written as \n",
    "\n",
    "```java\n",
    "Vec3 sumABC = a.add(b).add(c);\n",
    "```\n",
    "\n",
    "Now, consider the follwing declarations of vector variables `a` and `b`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "Vec3 a = new Vec3(0, 1, 0);\n",
    "Vec3 b = new Vec3(1, 0, 0);\n",
    "\n",
    "System.out.println(a);\n",
    "System.out.println(b);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume, the following calculations need to be performed:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\vec{c} & = & \\vec{a} + \\vec{b} \\\\\n",
    "\\vec{d} & = & \\vec{c} + \\vec{a}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Try to predict, which values variables `c` and `d` will have after the following implementation of this calculation has been executed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "Vec3 a = new Vec3(0, 1, 0);\n",
    "Vec3 b = new Vec3(1, 0, 0);\n",
    "Vec3 c = a.add(b);\n",
    "Vec3 d = c.add(a);\n",
    "\n",
    "System.out.println(c);\n",
    "System.out.println(d);"
   ]
  }
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