Solving system of 3 quadratic equations

[Gbl911]

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....

 

[anuttarasammyak]

$$\mathbf{v}\cdot\frac{\mathbf{P}-\mathbf{Q}}{|\mathbf{P}-\mathbf{Q}|}=1$$where
$$\mathbf{v}=\frac{\mathbf{V}}{R}$$
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+

 

[Gavran]

$r=((\vec{q}-\vec{p})/\|\vec{q}-\vec{p}\|)\cdot\vec{v}$
$r(\vec{q}-\vec{p})=\|\vec{q}-\vec{p}\|\vec{v}$
$(\vec{q}-\vec{p})/\|\vec{q}-\vec{p}\|=\vec{v}/r$

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.

 

[fresh_42]

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.

 

[Gavran]

The scalar product is not associative and the approach in the post #3 is wrong.
Sorry for that.