update ipynb

Kinematics of a Particle.ipynb

@@ -55,23 +55,13 @@
       "\n",
       "$$ (x,\\: y,\\: z) $$   \n",
       "\n",
-      "The position of a particle in space can also be represented by a vector in the Cartesian coordinate system, where the origin of the vector is at the origin of the coordinate system and the tip of the vector is at the point position:\n",
-      "\n",
-      "$$ \\mathbf{r} = \\mathbf{ x + y + z } $$\n",
-      "\n",
-      "Where now $ \\mathbf{ x,\\: y,\\: z } $ represent vectors in each axis direction of the coordinate system (note that the letters representing the vectors are shown in bold, another common convention is to use an arrow on top of the letter to represent a vector).    \n",
-      "The sum of the vectors above is different from the simple arithmetic sum. The arithmetics with vectors has to be performed with each component of the vectors.\n",
-      "\n",
-      "It's common to represent the vector with explicit mention to the axes directions using unit vectors $ \\mathbf{i,\\: j,\\: k} $:\n",
+      "The position of a particle in space can also be represented by a vector in the Cartesian coordinate system, with the origin of the vector at the origin of the coordinate system and the tip of the vector at the point position:\n",
       "\n",
       "$$ \\mathbf{r}(t) = x\\:\\mathbf{\\hat{i}} + y\\:\\mathbf{\\hat{j}} + z\\:\\mathbf{\\hat{k}} $$\n",
       "\n",
-      "A **unit vector** (or **versor**) is a vector whose length (or norm) is 1.   \n",
-      "The unit vector of a non-zero vector $\\mathbf{u}$ is the unit vector codirectional with $\\mathbf{u}$:\n",
-      "\n",
-      "$$ \\mathbf{\\hat{u}} = \\frac{\\mathbf{u}}{||\\mathbf{u}||} = \\frac{x\\:\\mathbf{\\hat{i}} + y\\:\\mathbf{\\hat{j}} + z\\:\\mathbf{\\hat{k}}}{\\sqrt{x^2+y^2+z^2}} $$\n",
+      "Where $ \\mathbf{i,\\: j,\\: k} $ are unit vectors in the directions of the axes $ \\mathbf{X, Y, Z} $. \n",
       "\n",
-      "In such case, $ x,\\: y,\\: z $ are now scalar quantities.\n",
+      "For a review on vectors, see [this notebook](http://nbviewer.ipython.org/urls/raw.github.com/duartexyz/BMC/master/Scalar%20and%20vector.ipynb).\n",
       "\n",
       "With this new notation, the coordinates of a point representing the position of a particle that vary with time would be expressed by the following position vector $ \\mathbf{r}(t)$:   \n",
       "\n",
@@ -83,9 +73,9 @@
       "\n",
       "And the unit vectors in each Cartesian coordinate in matrix form are given by:\n",
       "\n",
-      "$$ \\mathbf{\\hat{i}}= \\begin{bmatrix}1\\\\0\\\\0 \\end{bmatrix},\\; \\mathbf{\\hat{j}}=\\begin{bmatrix}0\\\\1\\\\0 \\end{bmatrix},\\; \\mathbf{\\hat{k}}=\\begin{bmatrix}0\\\\0\\\\1 \\end{bmatrix}$$\n",
+      "$$ \\mathbf{\\hat{i}}= \\begin{bmatrix}1\\\\0\\\\0 \\end{bmatrix},\\; \\mathbf{\\hat{j}}=\\begin{bmatrix}0\\\\1\\\\0 \\end{bmatrix},\\; \\mathbf{\\hat{k}}=\\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\end{bmatrix}$$\n",
       "\n",
-      "In [linear algebra](http://en.wikipedia.org/wiki/Linear_algebra), a set of unit linearly independent vectors (orthogonal in the Euclidean space) that can represent any vector via [linear combination](http://en.wikipedia.org/wiki/Linear_combination) is called a **basis**.   \n",
+      "In [linear algebra](http://en.wikipedia.org/wiki/Linear_algebra), a set of unit linearly independent vectors as the three vectors above (orthogonal in the Euclidean space) that can represent any vector via [linear combination](http://en.wikipedia.org/wiki/Linear_combination) is called a **basis**.   \n",
       "A basis is the foundation of creating a reference frame and we will study how to do that other time. "
      ]
     },
@@ -128,8 +118,8 @@
       "And in matrix form:\n",
       "\n",
       "$$ \\mathbf{v}(t) = \\begin{bmatrix}\n",
-      "\\frac{dx(t)}{dt} \\\\\\ \n",
-      "\\frac{dy(t)}{dt} \\\\\\\n",
+      "\\frac{dx(t)}{dt} \\\\\n",
+      "\\frac{dy(t)}{dt} \\\\\n",
       "\\frac{dz(t)}{dt}\n",
       "\\end{bmatrix}$$"
      ]
@@ -157,9 +147,9 @@
       "And in matrix form:\n",
       "\n",
       "$$ \\mathbf{a}(t) = \\begin{bmatrix}\n",
-      "\\frac{d^2x(t)}{dt^2} \\\\\\ \n",
-      "\\frac{d^2y(t)}{dt^2} \\\\\\\n",
-      "\\frac{d^2z(t)}{dt^2} \\\\\\\n",
+      "\\frac{d^2x(t)}{dt^2} \\\\\n",
+      "\\frac{d^2y(t)}{dt^2} \\\\\n",
+      "\\frac{d^2z(t)}{dt^2} \n",
       "\\end{bmatrix}$$"
      ]
     },
@@ -178,7 +168,7 @@
       "As the acceleration is the derivative of the velocity which is the derivative of position, the inverse mathematical operation is the [antiderivative](http://en.wikipedia.org/wiki/Antiderivative) (or integral):\n",
       "\n",
       "$$ \\begin{array}{l l}\n",
-      "\\mathbf{r}(t) = \\mathbf{r}_0 + \\int \\mathbf{v}(t) dt\\\\\\\n",
+      "\\mathbf{r}(t) = \\mathbf{r}_0 + \\int \\mathbf{v}(t) dt \\\\\n",
       "\\mathbf{v}(t) = \\mathbf{v}_0 + \\int \\mathbf{a}(t) dt \n",
       "\\end{array} $$"
      ]
@@ -198,24 +188,24 @@
       "#### Particle at rest\n",
       "\n",
       "$$ \\begin{array}{l l}\n",
-      "\\mathbf{a}(t) = 0 \\\\\\\n",
-      "\\mathbf{v}(t) = 0 \\\\\\\n",
+      "\\mathbf{a}(t) = 0 \\\\\n",
+      "\\mathbf{v}(t) = 0 \\\\\n",
       "\\mathbf{r}(t) = \\mathbf{r}_0\n",
       "\\end{array} $$\n",
       "\n",
       "#### Particle at constant speed\n",
       "\n",
       "$$ \\begin{array}{l l}\n",
-      "\\mathbf{a}(t) = 0 \\\\\\\n",
-      "\\mathbf{v}(t) = \\mathbf{v}_0 \\\\\\\n",
+      "\\mathbf{a}(t) = 0 \\\\\n",
+      "\\mathbf{v}(t) = \\mathbf{v}_0 \\\\\n",
       "\\mathbf{r}(t) = \\mathbf{r}_0 + \\mathbf{v}_0t\n",
       "\\end{array} $$\n",
       "\n",
       "#### Particle at constant acceleration\n",
       "\n",
       "$$ \\begin{array}{l l}\n",
-      "\\mathbf{a}(t) = \\mathbf{a}_0 \\\\\\\n",
-      "\\mathbf{v}(t) = \\mathbf{v}_0 + \\mathbf{a}_0t \\\\\\\n",
+      "\\mathbf{a}(t) = \\mathbf{a}_0 \\\\\n",
+      "\\mathbf{v}(t) = \\mathbf{v}_0 + \\mathbf{a}_0t \\\\\n",
       "\\mathbf{r}(t) = \\mathbf{r}_0 + \\mathbf{v}_0t + \\frac{\\mathbf{a}_0t^2}{2} \n",
       "\\end{array} $$"
      ]