Wednesday, October 15, 2014

Decomposition of Vector Advance Model

Decomposition of Vector (any mutually perpendicular) Advance Model




http://weelookang.blogspot.sg/2014/10/decomposition-of-vector-advance-model.html
author: lookang, Fu-Kwun, Andreu Glasmann, Wolfgang Christian, and Mario Belloni
CLICK https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_horizontalverticalsincos/horizontalverticalsincos_Simulation.xhtml
offline use:https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_horizontalverticalsincos.zip
source:https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_horizontalverticalsincos.zip
author of EJS 5: Paco.


Engage:

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. For example, the weight force 4 N downwards could be represented by the vector $ (A_{x'},A_{y'})$ = (4 cos($ \theta $),4sin($ \theta $)) (in 2 dimensions with the positive x' and y' axis) if the slope is making an angle of ( 90 -$ \theta $) with the x' axes (assume slope surface from left to right downwards) .

Component along the x' and y' direction:

As explained above a vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set. For simplicity, let's assume x and y direction as the reference axes, referring to the model above.
the equations for the components are:

 $ A_{x'} = A cos( \theta) $

 $ A_{y'} = A sin( \theta) $

Model:

This model uses the following equations:

Length or Magnitude |A|:

The length or magnitude or norm of the vector a is denoted by |A|.

 $ |A| = \sqrt {( A_{x'}^{2}+A_{y'}^{2})} $


Angle with positive x' axis:

The angle in radian of the vector that makes with the positive x axis is

$ \theta = tan^{-1} (\frac {A_{y'}}{A_{x'}}) $

the conversion to degree is

$ \theta_{degree} = (tan^{-1} (\frac{A_{y'}}{A_{x'}}) )( \frac{180}{\pi}) $

or

$ \theta_{degree} = ( \theta )( \frac{180}{\pi}) $

in the design of the slider values which shows in degree, the equation that converts back from degree to radian is used

$ \theta = ( \theta_{degree} )( \frac{\pi}{180}) $


Attribution:

when i made this model from a fresh ejss, the following works were reference from

  1. lookang, Fu-Kwun http://weelookang.blogspot.sg/2010/06/ejs-open-source-represent-vector.html
  2. Fu-Kwun, lookang  http://weelookang.blogspot.sg/2010/06/ejs-open-source-java-applet-resolving.html
  3. andreu glasmann, wolfgang christian, mario belloni Physlet Physics illustration 3.1. found in the Davidson College Digital Library for Physlets View the Physlet Two-Dimensional Kinematics Illustrations Package

Changes made:


  1. remade everything from lookang, Fu-Kwun http://weelookang.blogspot.sg/2010/06/ejs-open-source-represent-vector.html
  2. re-used some elements from andreu glasmann, wolfgang christian, mario belloni Physlet Physics illustration 3.1. found in the Davidson College Digital Library for Physlets View the Physlet Two-Dimensional Kinematics Illustrations Package to speed up development as per legally allowed by creative commons attribution, sharelike. i esepcially liked the ability to interact with the vector now, thanks to Prof Paco author of EJS 5.
  3. added ability to animate, thanks to idea from https://www.desmos.com/ and http://www.geogebra.org/
  4. added angle thanks to Prof Fu-Kwun guidance on some other works some years back!
  5. added ability to resolve along any perpendicular axes
  6. added ability to remember x' axes values after reset

Area of improvement:


  1. cannot figure out or cannot get the axis x' and y' to be drag-gable
  2. values to be shown on the view instead of the slider field

Example JavaScript Model Ill 3.1: Vector Decomposition written by Andreu Glasmann, Wolfgang Christian, and Mario Belloni


Other Good Model:

  1. https://www.geogebratube.org/student/m11066 
    https://www.geogebratube.org/student/m11066
    like: 1. ability to move vectors in screen, 2. values of components shown 5cos (60.16)
    dislike: 1. cannot fine control values example 60.16 instead of exactly 60