Problem: You are given 3 points in R^3. Find the fourth one.

[redivider]

Problem: You are given 3 points in R^3. Find the fourth one.

A(3,4,-1) B(1,0,-5) C(2,1,3)

|AT|=|BT|=sqt(14)

find T so that pyramid ABCT has volume V = 29/3

I have come to the conclusion that I have no idea how to solve this and I came here asking for help.

I have tried using the "V of the pyramid is 1/6 of mixed product AB(ACxAT)" because mixed product forms a paralelepiped or w/e its called in english, but I got a plane equation for some reason(edit: z=-x/2-y). I tried plotting the whole thing into blender to visualize, didnt help. Im lost. Help.

Edit: I managed to find *all 4 X coords, by doing all permutations of triplets of equations and dumping the ones that had complex solutions. Only 4 real remained. Now, I can get all 4 T points by getting the rest of the Y and Z coords for coresponding Xes, only "musculature" work remains as our teacher used to say. Check this link for really short summary on how to get X:

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* Im not sure if theres only 4 or more but I presume these are correct. Would still like an elegant solution more.

 

[stapel]

Problem: You are given 3 points in R^3. Find the fourth one.

A(3,4,-1) B(1,0,-5) C(2,1,3)

|AT|=|BT|=sqt(14)

find T so that pyramid ABCT has volume V = 29/3

I have tried using the "V of the pyramid is 1/6 of mixed product AB(ACxAT)"..., but I got a plane equation....

Please reply showing your efforts (even if you think they're wrong). Thank you!

 

[redivider]

Oh you wanna see my effort? Roger that;

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My effort is about 6 A4 papers, its all calculations because I am dealing with vectors in R^3 here. I have already told you what I intended to do but its not working out, I would appreciate If someone helped if no please refrain from replying. Im looking for help not flame. If I knew how to solve this I wouldnt be here. I can even show you some blender pictures or send you the .blend file but I fail to see how that helps in resolving the problem.

sincerely, alex

 

[redivider]

ps: I have found 1 solution by flipping over the 3x3 equation system which is T(4,2,-4) but I have no idea what I actually did. Moreover, judging by my blender renderings I believe there are 4 different Ts. Another X coordinate that I have also found is 812/233 but after flipping the 3x3 I keep getting complex solutions... As I said I cant picture the problem correctly and am therefore completely lost.