Appendix A - A Numerical Example on the Cross Product of 2 Tetrahedra

An Approach to the Four Body Problem

by Frank C. Fung - 1st published in December, 2007.

Top Of Page

Appendix A - A Numerical Example for the Cross-Product of 2 Tetrahedra

Brief Introduction :

In this Appendix , we shall do a numerical demonstration showing that , in the calculation of the { Angular Momentum Vector } :
- using the { Cross-Products } of the { Mass-Position Vectors } and the { Velocity Vectors } ,
- using the { Cross-Products } of the { Position Vectors } and the { Mass-Velocity Vectors } ,
- using the { Cross-Products } of the { RHO Vectors } and the { Mass-Velocity Vectors } ,
- using the { Cross-Products } of the { Mass-Position Vectors } and the { MU-Vectors } ;
will always all yield the exact same results , i.e. the one-and-the-same { angular momentum vector } .
This is done primarily to familiarize the reader with the concepts presented in { Section VII } and { Section VIII } .

go to Top Of Page

The { Position Vectors } and the { Mass-Position Vectors } :

Let us now bring back the 4 { Position Vectors } and the 4 { Mass-Position Vectors } we have used in { Section VII } :

And the data set here for the 2 sets of { vectors } are as follows :

Nodal Point	Position Vector			Mass in mass units	Mass-Position Vector
Nodal Point	X	Y	Z	Mass in mass units	X	Y	Z
Mass #1	+0.000	-225.000	-168.750	0.800	+0.000	-180.000	-135.000
Mass #2	+218.182	+0.000	-163.636	1.100	+240.000	+0.000	-180.000
Mass #3	-342.857	+257.143	-228.571	0.700	-240.000	+180.000	-160.000
Mass #4	+0.000	+0.000	+339.286	1.400	+0.000	+0.000	+475.000
Total	-124.675	+32.143	-221.671	4.000	+0.000	+0.000	+0.000
Mean	-31.169	+8.036	-55.418

We note here that the position of the { centroid } for the { Position-Vector Tetrahedron } is then at :
- X = -31.169 ,
- Y = -8.036 ,
- Z = -55.418 .

go to Top Of Page

The { Velocity Vectors } and the { Mass-Velocity Vectors } :

Let us now bring back the 4 { Velocity Vectors } and the 4 { Mass-Velocity Vectors } we have used in { Section VII } :

And the data set here for the 2 sets of { vectors } are as follows :

Nodal Point	Velocity Vector			Mass in mass units	Mass-Velocity Vector
Nodal Point	X	Y	Z	Mass in mass units	X	Y	Z
Mass #1	+300.000	+300.000	-225.000	0.800	+240.000	+240.000	-180.000
Mass #2	-218.182	+181.818	-54.545	1.100	-240.000	+200.000	-60.000
Mass #3	-228.571	-57.143	+285.714	0.700	-160.000	-40.000	+200.000
Mass #4	+114.286	-285.714	+28.571	1.400	+160.000	-400.000	+40.000
Total	-32.468	+138.961	-34.740	4.000	+0.000	+0.000	+0.000
Mean	-8.117	+34.740	+8.685

We note here that the position of the { centroid } for the { Veclocity-Vector Tetrahedron } is then at :
- X = -8.117 ,
- Y = +34.740 ,
- Z = +8.685 .

go to Top Of Page

Calculating the { Angular Momentum Vector } :

We then calculate the { Angular Momentum Vector } via the { Cross-Products } of :

the 4 { Mass-Position Vectors } and the 4 { Velocity Vectors } .

Nodal Point	Mass-Position Vector			Velocity Vector			Cross Product
Nodal Point	X	Y	Z	X	Y	Z	X	Y	Z
Mass #1	+0.000	-180.000	-135.000	+300.000	+300.000	-225.000	+81,000.000	-40,500.000	+54,000.000
Mass #2	+240.000	+0.000	-180.000	-218.182	+181.818	-54.545	+32,727.273	+52,363.636	+43,636.364
Mass #3	-240.000	+180.000	-160.000	-228.571	-57.143	+285.714	+42,285.714	+105,142.857	+54,857.143
Mass #4	+0.000	+0.000	+475.000	+114.286	-285.714	-28.571	+135,714.286	+54,285.714	+0.000
Total	+0.000	+0.000	+0.000	-32.468	+138.961	+34.740	+291,727.273	+171,292.208	+152,493.506

The { Angular Momentum Vector } calculated here is then characterized by :

X = +291,727.273 ,
Y = +171,292.208 ,
Z = +152,493.506 .

We can also calculate the { Angular Momentum Vector } via the { Cross-Products } of :

the 4 { Position Vectors } and the 4 { Mass-Velocity Vectors } .

Nodal Point	Position Vector			Mass-Velocity Vector			Cross Product
Nodal Point	X	Y	Z	X	Y	Z	X	Y	Z
Mass #1	+0.000	-225.000	-168.750	+240.000	+240.000	-180.000	+81,000.000	-40,500.000	+54,000.000
Mass #2	+218.182	+0.000	-163.636	-240.000	+200.000	-60.000	+32,727.273	+52,363.636	+43,636.364
Mass #3	-342.857	+257.143	-228.571	-160.000	-40.000	+200.000	+42,285.714	+105,142.857	+54,857.143
Mass #4	+0.000	+0.000	+339.286	+160.000	-400.000	+40.000	+135,714.286	+54,285.714	+0.000
Total	-124.675	+32.143	-221.672	+0.000	+0.000	+0.000	+291,727.273	+171,292.208	+152,493.506

The { Angular Momentum Vector } calculated here is then characterized by :

X = +291,727.273 ,
Y = +171,292.208 ,
Z = +152,493.506 .

the same as the one calculated immediately above .

go to Top Of Page

The { RHO Vectors } :

Let us recall that the { RHO Vectors } are the { centroid-based position vectors } for the { Position-Vector Tetrahedron } .

And the { RHO Vectors } are given by the formula :

[ RHO Vector ] = [ Position Vector ] minus the [ position vector for the centroid of the { Position-Vector Tetrahedron } ]

Nodal Point	Position Vector			Centroid of the Position-Vector Tetrahedron			RHO Vector
Nodal Point	X	Y	Z	X	Y	Z	X	Y	Z
Mass #1	+0.000	-225.000	-168.750	-31.169	+8.036	-55.418	+31.169	-233.036	-113.332
Mass #2	+218.182	+0.000	-163.636				+249.351	-8.036	-108.218
Mass #3	-342.857	+257.143	-228.571				-311.688	+249.107	-173.153
Mass #4	+0.000	+0.000	+339.286				+31.169	-8.036	+394.704
Total	-124.675	+32.143	-221.672				+0.000	+0.000	+0.000

We then calculate the { Angular Momentum Vector } via the { Cross-Products } of :

the 4 { RHO Vectors } and the 4 { Mass-Velocity Vectors } .

Nodal Point	RHO Vector			Mass-Velocity Vector			Cross Product
Nodal Point	X	Y	Z	X	Y	Z	X	Y	Z
Mass #1	+31.169	-233.036	-113.332	+240.000	+240.000	-180.000	+69,146.104	-21,589.286	+63,409.091
Mass #2	+249.351	-8.036	-108.218	-240.000	+200.000	-60.000	+22,125.812	+40,933.442	+47,941.558
Mass #3	-311.688	+249.107	-173.153	-160.000	-40.000	+200.000	+42,895.292	+90,042.208	+52,324.675
Mass #4	+31.169	-8.036	+394.704	+160.000	-400.000	+40.000	+157,560.065	+61,905.844	-11,181.818
Total	+0.000	+0.000	+0.000	+0.000	+0.000	+0.000	+291,727.273	+171,292.208	+152,493.506

The { Angular Momentum Vector } calculated here is then characterized by :

X = +291,727.273 ,
Y = +171,292.208 ,
Z = +152,493.506 .

same as the one's we have calculated above .

go to Top Of Page

The { MU Vectors } :

Let us recall that the { MU Vectors } are the { centroid-based position vectors } for the { Velocity-Vector Tetrahedron } .

And the { MU Vectors } are given by the formula :

[ MU Vector ] = [ Velocity Vector ] minus the [ position vector for the centroid of the { Velocity-Vector Tetrahedron } ]

Nodal Point	Velocity Vector			Centroid of the Velocity-Vector Tetrahedron			MU Vector
Nodal Point	X	Y	Z	X	Y	Z	X	Y	Z
Mass #1	+300.000	+300.000	-225.000	-8.117	+34.740	+8.685	+308.117	+265.260	-233.685
Mass #2	-218.182	+181.818	-54.545				-210.065	+147.078	-63.231
Mass #3	-228.571	-57.143	+285.714				-220.455	-91.883	+277.029
Mass #4	+114.286	-285.714	+28.571				+122.403	-320.455	+19.886
Total	-32.468	+138.961	+34.740				+0.000	+0.000	+0.000

We then calculate the { Angular Momentum Vector } via the { Cross-Products } of :

the 4 { Mass-Position Vectors } and the 4 { MU Vectors } .

Nodal Point	Mass-Position Vector			MU Vector			Cross Product
Nodal Point	X	Y	Z	X	Y	Z	X	Y	Z
Mass #1	+0.000	-180.000	-135.000	+308.117	+265.260	-233.685	+77,837.377	-41,595.779	+55,461.039
Mass #2	+240.000	+0.000	-180.000	-210.065	+147.078	-63.231	+26,474.026	+52,987.013	+35,298.701
Mass #3	-240.000	+180.000	-160.000	-220.455	-91.883	+277.029	+35,163.961	+101,759.740	+61,733.766
Mass #4	+0.000	+0.000	+475.000	+122.403	-320.455	+19.886	+152,215.909	+58,141.234	+0.000
Total	+0.000	+0.000	+0.000	+0.000	+0.000	+0.000	+291,727.273	+171,292.208	+152,493.506

The { Angular Momentum Vector } calculated here is then characterized by :

X = +291,727.273 ,
Y = +171,292.208 ,
Z = +154,493.506 .

same as the one's we have calculated above .

go to Top Of Page

This then finalizes our numerical demonstration here .

An Approach to the Four Body Problem

by Frank C. Fung - 1st published in December, 2007.

Top Of Page

Appendix A - A Numerical Example for the Cross-Product of 2 Tetrahedra

Brief Introduction :

go to Top Of Page

The { Position Vectors } and the { Mass-Position Vectors } :

go to Top Of Page

The { Velocity Vectors } and the { Mass-Velocity Vectors } :

go to Top Of Page

Calculating the { Angular Momentum Vector } :

go to Top Of Page

The { RHO Vectors } :

go to Top Of Page

The { MU Vectors } :

go to Top Of Page

go to Top Of Page

go back to the very last section : Section IX - Concluding Remarks

return to The Four Body Problem HomePage

original dated 2008-1-07