Solving system of 3 quadratic equations

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The discussion focuses on solving a system of three quadratic equations derived from the relationship between three-dimensional vectors Q, P, and V, where R is a scalar. The equations involve complex substitutions and lead to quadratic forms that are challenging to solve analytically. Participants note that numerical methods may be necessary due to the ill-conditioned nature of the equations, particularly when R is not a unit vector. The problem is likened to finding the intersection of three ellipsoids, emphasizing its complexity. Visualizing the equations in 3D space is suggested as a potential approach to understanding the intersections.
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Given the following equation:

R = ((Q-P) / |Q-P|) ⋅ V
where Q, P, and V are 3 dimensional vectors, R is a scalar, "" denotes the dot product, and |Q-P| is the magnitude of Q-P.

Assuming Q, V, and R are known and given 3 independent equations with different values for Q, V, and R that all correspond to one vector P, solve for all elements of P.

As an example the first equation would be:
R1 = ((Q1-P) / |Q1-P|) ⋅ V1

Expanding this out to view individual elements yields:
R1 = ((Q11-P1)V11 + (Q12-P2)V12 + (Q13-P3)V13) / sqrt((Q11-P1)2 + (Q12-P2)2+ (Q13-P3)2)

From here I have gotten some of the way through solving the 3 equations by substitution but it becomes unbearably long and requires the quadratic equation due to the sqrt... I have also tried using every numerical method I know to solve it but it seems the equation is far to ill conditioned and gradient methods are almost useless. I haven't found anything online regarding it and was hoping some of you who are good at math would be able to give me some guidance. I'm hoping there is a way without having to go through the substitution by hand but I may end up just doing that....
 
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\mathbf{v}\cdot\frac{\mathbf{P}-\mathbf{Q}}{|\mathbf{P}-\mathbf{Q}|}=1where
\mathbf{v}=\frac{\mathbf{V}}{R}
the equations are
\sum_{k=1}^3 v_{ik}(p_k-q_{ik})=\sqrt{\sum_{k=1}^3(p_k-q_{ik})^2}
or
(\sum_{k=1}^3 v_{ik}(p_k-q_{ik}))^2=\sum_{k=1}^3(p_k-q_{ik})^2
for i=1,2,3. By further calculation we can get three quadratic form equations of
\sum_{j,k=1}^3 a(i)_{jk}p_jp_k=0
when ##\mathbf{p}=(0,0,0)## is a solution. Otherwise
\sum_{j,k=1}^3 a(i)_{jk}p_jp_k=1
where i=1,2,3 and ##a(i)_{jk}=a(i)_{kj}##, symmetric.
I do not have enough knowledge to show how to solve it in general. Instead, I am interested in drawing figures of them in 3D space and observing the intersections and the crossing point if there are.

Example https://www.wolframalpha.com/input?i=plot+x^2+2y^2+3z^2+4xy+5yz+6zx=1+
 
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## r=((\vec{q}-\vec{p})/\|\vec{q}-\vec{p}\|)\cdot\vec{v} ##
## r(\vec{q}-\vec{p})=\|\vec{q}-\vec{p}\|\vec{v} ##
## (\vec{q}-\vec{p})/\|\vec{q}-\vec{p}\|=\vec{v}/r ##

In the Cartesian coordinate system there will be
## p_x=q_x-kv_x ##
## p_y=q_y-kv_y ##
## p_z=q_z-kv_z ##
where k is a non-zero real number.

The problem is unsolvable if ## \vec{v}/r ## is not a unit vector.
 
You have basically three different ellipsoids and ask for their intersection. This is not a trivial problem. I assume that only numerical methods can help here.
 
The scalar product is not associative and the approach in the post #3 is wrong.
Sorry for that.
 

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