Another linear algebra problem

[Steven G]

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.

 

[pka]

View attachment 14989

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.

 

[Romsek]

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 $\displaystyle x$

 

[Steven G]

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 $\displaystyle (\rho_1)^2- (\rho_2)^2, (\rho_2)^2- (\rho_3)^2 \ \& \ (\rho_3)^2- (\rho_4)^2$