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
$$\sum_{k=1}^3 v_{ik}(p_k-q_{ik})=\sqrt{\sum_{k=1}^3(p_k-q_{ik})^2}$$
for i=1,2,3. With $\{q_{ik}\}$ and $\{v_{ik}\}$ given, you expect to decide $(p_1,p_2,p_3)$ as cross point of three 3D quadratic surfaces.