Another linear algebra problem

Steven G

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Dec 30, 2014
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I tried a few ideas and I assure that nothing helped. Any hints/sols will be helpful (that will get me back to help on the forum faster)
The problem is below. It actually looks looks easy especially with the hint. Oh well.
888.jpg
 
After having read this several times, I really have very little idea what it means (I taught vector analysis courses for over thirty years) But I have never
this notation. Here is my guess: thinking of \(\displaystyle x~\&~a_k\) as points then \(\displaystyle x-a_k\) is a vector from \(\displaystyle a_k\) to \(\displaystyle x\) the length of which is \(\displaystyle \|x-a_k\|=\rho_k\).
We know the \(\displaystyle (\rho_k)^2=(x-a_k)\cdot(x-a_k)=(x\cdot x)+2(x\cdot a_k)+(a_k\cdot a_k)=\|x\|^2+2(x\cdot a_k) +\|a_k\|^2\) [those are dot products]
Let me reiterate, I guessing. Please look at that and tell me what else you know about the notation. If you have a textbook I may know more.
 
This is a typical active radar or sonar triangulation problem.

In this problem there are 4 radar stations at point \(\displaystyle a_1,~\dots,~a_4\)

\(\displaystyle \rho_i^2 = (x-a_i)\cdot (x-a_i)\)

you can expand all this out using \(\displaystyle x = (x,y,z),~a_i = (a_{ix}, a_{iy}, a_{iz})\)

and then use the hint to coalesce it into matrix equations on the coordinates of \(\displaystyle x\)
 
After having read this several times, I really have very little idea what it means (I taught vector analysis courses for over thirty years) But I have never
this notation. Here is my guess: thinking of \(\displaystyle x~\&~a_k\) as points then \(\displaystyle x-a_k\) is a vector from \(\displaystyle a_k\) to \(\displaystyle x\) the length of which is \(\displaystyle \|x-a_k\|=\rho_k\).
We know the \(\displaystyle (\rho_k)^2=(x-a_k)\cdot(x-a_k)=(x\cdot x)+2(x\cdot a_k)+(a_k\cdot a_k)=\|x\|^2+2(x\cdot a_k) +\|a_k\|^2\) [those are dot products]
Let me reiterate, I guessing. Please look at that and tell me what else you know about the notation. If you have a textbook I may know more.
I looked at at the textbook and it seems that what you were doing is correct (what else could you do--I did this myself as well)
Compute the difference for \(\displaystyle (\rho_k)^2 and (\rho_m)^2\). Then the three linear equations will be the results of \(\displaystyle (\rho_1)^2- (\rho_2)^2, (\rho_2)^2- (\rho_3)^2 \ \& \ (\rho_3)^2- (\rho_4)^2\)
 
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