Dear Majid Zohrehbandian

((I know that the 3 vectors x,y,z in Rn , where angles between them are 120`are coplanar . ))

In general, the above statement is wrong.

Regards

Dear Dr. Issam Kaddoura ,

You say that the 3 vectors with angles 120` between them may not CoPlanar and they can produce a Tetrahedron?

Can you explain or give me an example?

Then, the 6 vectors in my question are not CoPlanar.

Dear Majid Zohrehbandian

v1 = (1, 0, 0)

v2= (-1/2, 1, 0)

v3= (-1/2, -3/4, 1)

Best regards

But it's not true.

Here -1/2 = V1.V2 = |V1||V2| cos(V1oV2) = 1 x sqrt(5)/2 x cos(V1oV2)

Then, cos(V1oV2) = - sqrt(5)/5 , and deg(V1oV2) = 116.56' .

Dear Mehdi Mesrizadeh

It's not so easy as you think. Because, in addition to the mentioned equations you should consider conditions about the normality of the vectors.

About the 3 vectors, I think they are coplanar and I think I could proved it.

(see my question and answer in ResearchGate:

https://www.researchgate.net/post/Isnt_there_any_Tetrahedron_with_3_vertex_angles_a_b_a_b_ie_is_it_true_that_3_vectors_in_Rn_where_angles_between_them_are_a_b_a_b_are_coplanar

).

But about 6 vectors in this question, I'm not sure. I can propose a solution approach as I did for 3 vectors, but here by moving a vector in form of normal to the first plane, the transformed vector should remain on the bisector of 2 adjacent vector.

However, I am grateful if you can produce a counter example for each of these questions (question about 3 vectors and question about 6 vectors). Continuation of my project and my conjecture to solve it, depends on this coplanarity of these 6 vectors.

Regards

Dear Majid Zohrehbandian

Indeed it is an interesting problem. It is more profound than it seems. Your note about my counterexample is true. I have proved that your claim is valid in R3, but I am not sure about higher dimensions.

Regards

Dear Issam Kaddoura ,

Me too. As you said, I'm not sure about vectors in Rn. But I think my proof may be true for 3 vectors on Rn.

Regards

My proof about the question is as follows:

I have used these theorems to solve the Vertex Cover Problem and based on solving a semidefinite programming relaxation.

I hope it is correct. See:

https://zenodo.org/record/4312668

Isn't this a counterexample in R⁴:

v_1 = (1,0,0,0)

v_2 = (-1/2,sqrt(3)/2,0,0)

v_3 = (-1/2,-sqrt(3)/2,0,0)

v_23 = (0,0,1,0)

v_13 = (0,0,-1/2,sqrt(3)/2)

v_12 = (0,0,-1/2,-sqrt(3)/2)

Dear Christian Bär ,

You are right and what you mentioned is a counterexample for my first question.

But, I in last answer which I myself proved the problem, I said that the vectors are coplannar OR V_i . V_ij =0.

Your example has the second property and 3 vectors V_i are perpendicular to 3 vectors V_ij.

I made a mistake in computation of t in last proof of Theorem 2 and maybe Theorem 2 is incorrect.