Statics in 3D, there must a be faster way to do this.

[goonking]

Given this, is there a faster way to compute this or must I do 3 separate 3x3 determinants?

I can probably use cross product of each term, for example , (2rk) X (A_xi) = (-2rA_x j) (not forgetting the negative sign for j)

next, (2rk) X (A_yj) = (2rA_yi)

and so on... but that feels too slow as well and can get very messy.

 

[PhanthomJay]

Break up each of the forces into their x y and z components and sum them separately equal to 0. Then for the sum of moments equals 0 for each force component about an axis, it's

$M_x = F_yz + F_zy$

$M_y = F_xz + F_zx$

$M_z = F_xy + F_yx$

where x y and z are the perpendicular distances from the line of action of the component force to the axis about which moments are being summed.
I don't know if that's easier for you, but it avoids determinants and vector math equations .

 

[DanielSauza]

Divide your 3D problems into 2D problems. Just remember, the forces and moments in a plane must be in equilibrium.