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    "#### 1. Derive expressions for $\\hat{r}$ and $\\hat{\\theta}$ as a function of $\\hat{i}$ and $\\hat{j}$"
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    "Ideally, in a cartesian frame of reference, a resultant unit vector $\\hat{r}$, should be in terms of the cartesian unit vectors $\\hat{i}$ and $\\hat{j}$ as follows: $$\\hat{r} = \\hat{i} +\\hat{j}$$\n",
    "\n",
    "Let's assume for a second that we rotate our coordinate system counter-clockwise an arbitrary angle $\\theta$, such that the coordinate system is aligned with the direction of the unit vector $\\hat{r}$. As a result, our angle $\\hat{r}$ can be described as \"components\" of the unit vectors. Therefore our last expression would change in the follwoing form:$$\\hat{r} = \\cos{(\\theta)}\\hat{i} +\\sin{(\\theta)}\\hat{j}$$\n",
    "\n",
    "For instance, given that the unit vector $\\hat{r}$ is orthonormal (makes a $\\pi/2$ angle) in the same coordinate system) to the unit vector $\\hat{\\theta}$, it can be written as follows: $$\\hat{\\theta} = \\sin{(\\theta)}\\hat{i} +\\cos{(\\theta)}\\hat{j}$$. Notice that the first term could have a negative sign, depending on the rotation of our system. \n",
    "\n",
    "Also notice that if these expressions are correct, the scalar profuct of $\\hat{r}$ and $\\hat{\\theta}$ should be zero, since both are orthonormal.  \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. Since the two frames are equialent (you should choose one over the other based on the convenience) they have the same number of basis vectors obeying the same properties. What property must be satisfied by $\\{\\hat{r},\\hat{\\theta}\\}$ as a set?"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initially, polar coordinates are more convenient when circular motion is involved. \n",
    "Referencing our expressions on last problem, when the angle $\\theta = \\{0,\\frac{\\pi}{2},\\pi,\\frac{3}{\\pi},2\\pi\\}$, it is more convenient to use cartesian coordinates, since there is no need to rotate our system in the same domain of cartesian coordinates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3. What is one essential porperty that the set of basis vectors$\\{\\hat{r},\\hat{\\theta}\\}$, $\\{\\hat{i},\\hat{j}\\}$ do not share? *Hint: Thank about describing a trajectory as a function of time using both coordinate systems*"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finding the derviative of both functions, the velocity of a punctual mass in the plance described in cartesian coodinates describes the motion in terms of linear velocities in both axis. \n",
    "For the case of polar coordinates, the velocity of the object is described in terms of the angular velocity, since includes the derviative of the angle with respect to time. This property makes polar coordinates more suitable for circular motion."
   ]
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