java - how to calculate plane equation using three points and two vectors? -

i trying calculate 2 plane equations data of 2 dicom images , intersection line these 2 plane equations. dicom tag imagepositionpatient has 3 coordinates x,y,z , tag imageorientationpatient has 2 vectors each 3 components. how equation of plane can calculated using these 3 coordinates , 2 vectors?

this code:

public void showlocalizerline(view v) {     //getting values     infooffirst();  //this gets img position , orientation of src img     infoofsecond(); //this gets img position , orientation of dst img     string[] src_position_array = new string[3];     src_position_array = firstimg_position.split("\\\\");     src_position_xo = float.parsefloat(src_position_array[0]);     src_position_yo = float.parsefloat(src_position_array[1]);     src_position_zo = float.parsefloat(src_position_array[2]);      string[] src_orientation_array = new string[6];     src_orientation_array = firstimg_orientation.split("\\\\");     src_orientation_vector1_a = float.parsefloat(src_orientation_array[0]);     src_orientation_vector1_b = float.parsefloat(src_orientation_array[1]);     src_orientation_vector1_c = float.parsefloat(src_orientation_array[2]);     src_orientation_vector2_d = float.parsefloat(src_orientation_array[3]);     src_orientation_vector2_e = float.parsefloat(src_orientation_array[4]);     src_orientation_vector2_f = float.parsefloat(src_orientation_array[5]);      string[] dst_position_array = new string[3];     dst_position_array = secimg_position.split("\\\\");     dst_position_xo = float.parsefloat(dst_position_array[0]);     dst_position_yo = float.parsefloat(dst_position_array[1]);     dst_position_zo = float.parsefloat(dst_position_array[2]);      string[] dst_orientation_array = new string[6];     dst_orientation_array = secimg_orientation.split("\\\\");     dst_orientation_vector1_a = float.parsefloat(dst_orientation_array[0]);     dst_orientation_vector1_b = float.parsefloat(dst_orientation_array[1]);     dst_orientation_vector1_c = float.parsefloat(dst_orientation_array[2]);     dst_orientation_vector2_d = float.parsefloat(dst_orientation_array[3]);     dst_orientation_vector2_e = float.parsefloat(dst_orientation_array[4]);     dst_orientation_vector2_f = float.parsefloat(dst_orientation_array[5]);      //calculations     float dst_nrm_dircos_x = dst_orientation_vector1_b*dst_orientation_vector2_f - dst_orientation_vector1_c*dst_orientation_vector2_e;     float dst_nrm_dircos_y = dst_orientation_vector1_c*dst_orientation_vector2_d - dst_orientation_vector1_a*dst_orientation_vector2_f;     float dst_nrm_dircos_z = dst_orientation_vector1_a*dst_orientation_vector2_e - dst_orientation_vector1_b*dst_orientation_vector2_d;      float src_pos_x = src_position_xo - dst_position_xo;     float src_pos_y = src_position_yo - dst_position_yo;     float src_pos_z = src_position_zo - dst_position_zo;     float dst_pos_x = dst_orientation_vector1_a*src_pos_x + dst_orientation_vector1_b*src_pos_y + dst_orientation_vector1_c*src_pos_z;     float dst_pos_y = dst_orientation_vector2_d*src_pos_x + dst_orientation_vector2_e*src_pos_y + dst_orientation_vector2_f*src_pos_z;     float dst_pos_z = dst_nrm_dircos_x*src_pos_x + dst_nrm_dircos_y*src_pos_y + dst_nrm_dircos_z*src_pos_z;      } 

from class names guess imagepositionpatient gives point (say a) on plane , imageorientationpatient (say n) gives normal. (i wouldn't know correct since told me 3d vectors have in mind).

then plane equation given dot(n, a) = dot(n, x), x position variable , n normalized. gives ax + + cz = d cartesian equation.