eliminate 3 variables in a 6-variable three simultaneous quadratic equations

Abdulhafeeth

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Please help me how I could eliminate 3 variables in the following simultaneous equations so that they become in 3 variables instead of 6:

eliminate 3 variables in a 9-variable three simultaneous quadratic equations.png
 
[MATH]x_1^2 + y_1^2 + z_1^2 = c_1.[/MATH]
[MATH]x_2^2 + y_2^2 + z_2^2 = c_2.[/MATH]
[MATH](x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 = c_3 \implies[/MATH]
[MATH]x_1^2 - 2x_1x_2 + x_2^2 + y_1^2 - 2y_1y_2 + y_2^2 + z_1^2 - 2z_1z_2 + z_2^2 = c_3 \implies[/MATH]
[MATH](x_1^2 + y_1^2 + z_1^2) + (x_2^2 + y_2^2 + z_2^2) - 2(x_1x_2 + y_1y_2 + z_1z_2) = c_3 \implies[/MATH]
[MATH]c_1 + c_2 - 2(x_1x_2 + y_1y_2 + z_1z_2) = c_3 \implies x_1x_2 + y_1y_2 + z_1z_2 = -\ \dfrac{c_3 - c_2 - c_1}{2}.[/MATH]
[MATH]\text {Let } c = -\ \dfrac{c_3 - c_2 - c_1}{2},\ p = 2x_1x_2,\ q = y_1y_2, \text { and } r = z_1z_2.[/MATH]
[MATH]\therefore p + q + r = c.[/MATH]
Is that helpful?
 
I appreciate your answer very much, but still there are six variables!
In fact,these variables are actually the coordinates of a body in space: (x1,y1,z1),(x2,y2,z2)
 
I appreciate your answer very much, but still there are six variables!
In fact,these variables are actually the coordinates of a body in space: (x1,y1,z1),(x2,y2,z2)
The response above is showing you a way (first step) to convert your equations with six variables to an equation with three variables (p, q & r).

Now it is your turn to show us how you would continue here....
 
I appreciate your answer very much, but still there are six variables!
In fact,these variables are actually the coordinates of a body in space: (x1,y1,z1),(x2,y2,z2)
It would help a lot if you put the question into its context. WHY do you want to eliminate three variables? WHICH should they be, if it matters? And if, as it appears, you showed us one step in some larger explanation of something, what did they do next?
Please help me how I could eliminate 3 variables in the following simultaneous equations so that they become in 3 variables instead of 6:

View attachment 15024
I do see that in saying "eliminate 3 variables" (which JeffM did, reducing to three other variables), you misquoted the problem, which explicitly says to "eliminate 3 coordinates", implying that changing to new variables is not what they have in mind. But we don't know why the former is not acceptable, because we don't know the goal.

EDIT: After writing that, I searched for a phrase from the image, and found it appears to be from this book:


Nothing is shown of how they eliminate coordinates, but the goal is simply to show that there are three degrees of freedom. Doesn't what JeffM did accomplish that goal?
 
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Yes,you're right,Dr.Peterson,I should've put my question clearly;I think the author of the book (which is the one you mentioned) meant eliminating any three of the six coordinates.But what JeffM really did is just an operation of masking rather than eliminating.

Anyway,I'm happy with the communication I've got here,and I'd like to thank you for your time and care.
 
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