I have six unit vectors in real $3$D space
v1,v2,v3,v4,v5,v6;Represent the dot product between some of the vectors as the Cos of the angle between them
v1.v2=Cos[Subscript[\[Theta], 1, 2]];v1.v3=Cos[Subscript[\[Theta], 1, 3]];v3.v2=Cos[Subscript[\[Theta], 3, 2]];v1.v4=Cos[Subscript[\[Theta], 1, 4]];v2.v5=Cos[Subscript[\[Theta], 2, 5]];v3.v6=Cos[Subscript[\[Theta], 3, 6]];We also know the exact values of the following dot products;
v4.v5= -1/3;v4.v6= -1/3;v6.v5= -1/3;Since we had six vectors there should be in total $^6C_2=15$ angles between them. Whereas we have only defined $9$ of them. I need to represent the remaining $6$ angles in terms of the \[Theta] parameters defined above.
One way to approach would be to represent the first three in terms of the last three,
v1= a v4 + b v5 + c v6;v2= d v4 + e v5 + f v6;v3= g v4 + h v5 + i v6;Taking the dot product of these three vectors with every other vector would give us a system of equations which can be used to solve the remaining angles. But how do I do it?
EDITAfter applying the substitution for the first three vectors mentioned above, we have the following set of equations,
a-b/3-c/3=t14;-d/3+e-f/3=t25;-g/3-h/3+i=t36;a*d+b*e+c*f-a*e/3-a*f/3-b*d/3-b*f/3-c*d/3-c*e/3=t12;a*g+b*h+c*i-a*h/3-a*i/3-b*g/3-b*i/3-c*g/3-c*h/3=t13;g*d+h*e+i*f-g*e/3-g*f/3-h*d/3-h*f/3-i*d/3-i*e/3=t32;a^2+b^2+c^2-2/3(a*b+a*c+b*c)=1;d^2+e^2+f^2-2/3(d*e+d*f+e*f)=1;g^2+h^2+i^2-2/3(g*h+g*i+h*i)=1;Given these constraints, I need to represent the following quantitites,
-a/3+b-c/3;-a/3-b/3+c;d-e/3-f/3;-d/3-e/3+f;g-h/3-i/3;-g/3+h-i/3;How do I solve for linear combination for variables in mathematica?