Coordinate axes

Examples of Coordinate axes in the following topics:

• Adding and Subtracting Vectors Graphically

  • Vectors may be added or subtracted graphically by laying them end to end on a set of axes.
  • The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes.
  • To start, draw a set of coordinate axes.
  • Next, draw out the first vector with its tail (base) at the origin of the coordinate axes.
  • The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes.

• Scalars vs. Vectors

  • The two parts are its length which represents the magnitude and its direction with respect to some set of coordinate axes.
  • Typically this reference point is a set of coordinate axes like the x-y plane.

• Origin of Pressure

  • When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall ($\Delta p$) is:
  • (This does not mean that each particle always travel in 45 degrees to the coordinate axes. )

• Components of a Vector

  • Vectors are geometric representations of magnitude and direction which are often represented by straight arrows, starting at one point on a coordinate axis and ending at a different point.
  • A vector is defined by its magnitude and its orientation with respect to a set of coordinates.
  • To visualize the process of decomposing a vector into its components, begin by drawing the vector from the origin of a set of coordinates.
  • The original vector, defined relative to a set of axes.
  • The horizontal component stretches from the start of the vector to its furthest x-coordinate.

• Elastic Collisions in Multiple Dimensions

  • To solve a two dimensional elastic collision problem, decompose the velocity components of the masses along perpendicular axes.
  • The general approach to solving a two dimensional elastic collision problem is to choose a coordinate system in which the velocity components of the masses can be decomposed along perpendicular axes .

• Introduction to Spherical and Cylindrical Harmonics

  • In this section we will apply separation of variables to Laplace's equation in spherical and cylindrical coordinates.
  • Spherical coordinates are important when treating problems with spherical or nearly-spherical symmetry.
  • For instance, in Cartesian coordinates the surface of the unit cube can be represented by:
  • On the other hand, if we tried to use Cartesian coordinates to solve a boundary value problem on a spherical domain, we couldn't represent this as a fixed value of any of the coordinates.
  • This equation can be integrated to give: $\phi(x)=ax+b$.

• Problems

  • This problem will work best if you have a sheet of graph paper.In a spacetime diagram one draws a particular coordinate (in our case $x$) along the horizontal direction and the time coordinate vertically.People also generally draw the path of a light ray at 45$^\circ$.This sets the relative units of the two axes.
  • Draw a spacetime diagram and label the axes $x$ and $t$.The $t$-axis is the path of Emma through the spacetime.

• B.4 Chapter 4

  • In a spacetime diagram one draws a particular coordinate (in our case $x$) along the horizontal direction and the time coordinate vertically.
  • This sets the relative units of the two axes.
  • Draw a spacetime diagram and label the axes $x$ and $t$.
  • The light ray bisects the angle between the $x$ and $t$ axes.

• Tensors

  • Right now, we can build a contravariant vector by taking a set of coordinates $x^i$ for a event in spacetime and we can construct a covariant vector by applying the metric $\eta_{\mu\nu}$ to lower the index of the vector.
  • $\displaystyle E_x = - {\phi}{x} -\frac{1}{c} {A_x}{t} = A_{0,1} - A_{1,0} \\ B_x = {A_z}{y} - {A_y}{z} = A_{3,2} - A_{2,3}$
  • We also need to use the proper time $\tau$ instead of the coordinate time $t$, this gives

• Constant Acceleration

  • Analyzing two-dimensional projectile motion is done by breaking it into two motions: along the horizontal and vertical axes.
  • The most important fact to remember is that motion along perpendicular axes are independent and thus can be analyzed separately.
  • Because the acceleration due to gravity is along the vertical direction only, $a_x = 0$.
  • We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes.
  • The horizontal motion is simple, because $a_x = 0$ and $v_x$ is thus constant.