Finding position and orientation of a Frame on another frame in 3D Space

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I am trying to figure out how to fine the position and orientation of a frame (Frame_B) in another frame (Frame_A)

I have this data:

  • 3 points (P1_A, P2_A, P3_A) in Frame_A, these points define Frame_B plane xy.
  • The vector P1_A, P2_A is parallel to X axis of Frame_B.
  • The vector P1_A, P3_A is parallel to Y axis of Frame_B. ***
  • 3 points (P4_A, P5_A, P6_A) in Frame_A, these points define Frame_B plane xz.
  • 3 points (P7_A, P8_A, P9_A) in Frame_A, these points define Frame_B plane yz.

As you can see, I am not a mathematician, so please be gentle.

Thank you very much

Edit: *** I made a mistake in my data:

The vector P1_A, P3_A is parallel to Y axis of Frame_B. not to Z axis of Frame_B.

Here are same detailed data, please see images below Image with different points

Expected position and orientation

Edit2: I managed to find the Frame_B origin coordinates by resolving the 3 planes equations system. But I have no idea in how to find the orientations. Could anyone please help? Thank you in advance