Step 12 - The "N" Matrix

	3563.8925 km	5052.4587 km	6016.5229 km
r =	-346.24563 km	1056.2713 km	2392.5540 km
	6251.1122 km	5011.6366 km	3143.8252 km

Variable	Symbol	Value
First Geocentric Range	r₁	7,204.0002 km
Second Geocentric Range	r₂	7,194.4110 km
Third Geocentric Range	r₃	7,197.6732 km

	-8,338,127.7 km²	-8,669,878.9 km²	16,044,659 km²
Z =	13,722,552 km²	14,268,579 km²	-26,405,705 km²
	5,513,829.1 km²	5,733,199.8 km²	-10,610,000 km²

Another intermediate matrix will be required before we can finally determine the velocity vector of the satellite. The intermediate "N Matrix" is determined by using the following equations:

N₁₁ = r₁Z₁₂ + r₂Z₁₃ + r₃Z₁₁N₂₁ = r₁Z₂₂ + r₂Z₂₃ + r₃Z₂₁N₃₁ = r₁Z₃₂ + r₂Z₃₃ + r₃Z₃₁

N₁₂ = Z₁₁ + Z₁₂ + Z₁₃N₂₂ = Z₂₁ + Z₂₂ + Z₂₃N₃₂ = Z₃₁ + Z₃₂ + Z₃₃

N₁₃ = (r₂-r₃)r₁₁ + (r₃-r₁)r₁₂ + (r₁-r₂)r₁₃N₂₃ = (r₂-r₃)r₂₁ + (r₃-r₁)r₂₂ + (r₁-r₂)r₂₃N₃₃ = (r₂-r₃)r₃₁ + (r₃-r₁)r₃₂ + (r₁-r₂)r₃₃

	-7.0410564 x 10⁹ km³	-963,347.6 km²	14,100.605 km²
N =	1.1587796 x 10¹⁰ km³	1,585,426 km²	17,389.173 km²
	4.6560117 x 10⁹ km³	637,028.9 km²	-21,954.234 km²

We can also define two values using this matrix that will be used in the next step:

N₁ = [ (N₁₁)² + (N₂₁)² + (N₃₁)² ]^1/2N₂ = [ (N₁₂)² + (N₂₂)² + (N₃₂)² ]^1/2

N₁ = 1.4336385 x 10¹⁰ km³
N₂ = 1,961,484.1 km²

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Step 12: N Matrix Was Last Modified On September 23, 2013