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tokens: background magnetic field than those observed in the synthetic observations (slight preference of perpendicular alignment; Fig. 6 ), which were generated from a trans- to super-Alfvénic simulation. Combined with other polarimetric properties, we suggest that the relative alignment between dense cores and pc-scale magnetic fields depends on the magnetic properties of the cloud, and could potentially provide an alternative approach than the traditional Zeeman observations on investigating the magnetization level of the cloud. APPENDIX A: CORES IDENTIFIED IN GAS DATA AND THEIR RELATIVE ALIGNMENTS Figs A1 – A3 show all subregions observed by GAS overlapped with the core boundaries identified by astrodendro . The relative alignment between cores and background magnetic field (i.e. the angle between core major axis and background magnetic field direction) in each subregion is plotted in Fig. A4 . Figure A1. Cores (red contours) identified in each individual region in the Ophiuchus cloud using NH 3 integrated emission (colour maps, in K km s −1 ) observed by GAS. Note that cores that are more rounded (with minor/major ratio >0.8) are removed from the analysis and not shown here. Dendrogram leaves without obvious central peak emission are also excluded. Figure A2. Same as Fig. A1 , but for the Perseus cloud. Figure A3. Same as Fig. A1 , but for the Taurus cloud. Figure A4. The cumulative distributions of the core–background magnetic field alignment for the subregions within the three clouds: Ophiuchus (left), Perseus (middle), and Taurus (right). APPENDIX B: RELATIVE ANGLES IN 2D AND 3D Here, we briefly describe the difference between measuring relative angle between two random vectors in 3D and 2D spaces. Without loss of generality, we assume both vectors are unit vectors and one is pointing straight up (+ z for 3D case and + y for 2D case). The probability for the second vector to have an angle θ with respect to the first one is therefore proportional to the surface area of the stripe of constant latitude θ on the 3D sphere (), or the arc length at θ of the unit circle ( R dθ). For unit vectors, R = 1, and since we only consider angles within [0, 90] deg, the cumulative probability of the relative angle to be smaller than θ is therefore Note that the probability is normalized by the total surface area of the sphere (in 3D) or the circumference of the whole circle (in 2D). This explains why a set of 3D random vectors will show a linear distribution in cos α instead of α, where α is the relative angle between any two vectors from the set. Equations ( B1 ) and ( B2 ) are illustrated in Fig. B1 as black dotted lines, which also shows that two vectors in 3D are more likely to be perpendicular than parallel if measured in degrees. Figure B1. Comparing the relative angles in 3D (top) and 2D (bottom), for both analytic solutions of equations ( B1 ) and ( B2 ) (black dotted lines) and between two sets of random vectors (solid lines), showing the angle in