Hello, I have a simple(or not?) math problem:(adsbygoogle = window.adsbygoogle || []).push({});

I have equations of 3 lines in R3 trought the origin:

line l_{1}:

|λ_{1}*x+β_{1}*y+γ_{1}*z=0

|λ_{2}*x+β_{2}*y+γ_{2}*z=0

line l_{2}:

|λ_{3}*x+β_{2}*y+γ_{3}*z=0

|λ_{4}*x+β_{3}*y+γ_{4}*z=0

line l_{3}:

|λ_{5}*x+β_{4}*y+γ_{5}*z=0

|λ_{6}*x+β_{5}*y+γ_{6}*z=0

I know every λ, β and γ - they are real constants.

I also have a plane δ and :

l_{1}intersects δ in point p_{1},

l_{2}intersects δ in point p_{2},

l_{3}intersects δ in point p_{3}

I also know:

....the distance between p_{1}and p_{2}= h

....the distance between p_{2}and p_{3}= w

....p_{1}, p_{2}and p_{3}form a right triangle with right angle at p_{2}(h^{2}+w^{2}=(p1p3)^{2})

I want to find the equation of δ in terms of λ, β, γ, h and w

_______________________________

it should be easy to find the equation of the plane if I find the points p_{1}, p_{2}and p_{3}

I think I should compose a system, containing:

....the first six equations(which are linear, so it should be easy to solve with matrices),

....the equation for right angle in R3: w^{2}+h^{2}=(p_{1}p_{2})^{2},

....the two equations for distances w and h: h^{2}=(p_{1x}-p_{2x})^{2}+(p_{1y}-p_{2y})^{2}+(p_{1z}-p_{2z})^{2}and w^{2}=(p_{2x}-p_{3x})^{2}+(p_{2y}-p_{3y})^{2}+(p_{2z}-p_{3z})^{2}

There are 9 equations with 9 unknowns (3 points, each with 3 coordinates) - it should be solvable.

Have you got any ideas how to solve this? If there weren't 3 quadratic equations it would be easy.

*I know that there should be 2 planes, matching the conditions - one that I've drawn(see the image) and the other is at the opposite side of the origin. I think the two roots of the quadratic equations should find them.

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# A simple camera problem

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