Equations for shoulder external rotation angle calculation

Points (x1, y1z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4) represent marker positions of acromion, elbow, wrist, and Th8, respectively, in the 3D coordinate system. Inner product A(X1, Y1, Z1) was projected perpendicularly from established triangle between acromion, elbow, and Th8 markers.

Vector A (X1,Y1, Z1) was calculated by the equations below:
X1 = (y2-y1)*(z3-z1)-(y3-y1)*(z2-z1)
Y1 = (z2-z1)*(x3-x1)-(z3-z1)*(x2-x1)
Z1 = (x2-x1)*(y3-y1)-(x3-x1)*(y2-y1)

Inner product B (X2, Y2, Z2) was calculated with acromion, elbow, and Th8 markers by the equations below:
X2 = (y2-y1)*(z4-z1)-(y4-y1)*(z2-z1)
Y2 = (z2-z1)*(x4-x1)-(z4-z1)*(x2-x1)
Z2 = (x2-x1)*(y4-y1)-(x4-x1)*(y2-y1)

Shoulder external rotation was defined as the inner product between the two vectors:

The angle between two vectors (A,B) was obtained by calculating acosθ, which was defined as shoulder external rotation angle in this study.