Ultrasonix linear probe calibration matrix


I have a little confusion about the calibration matrix (referred to as C matrix) for the linear probe. As written in this website: http://www.ultrasonix.com/wikisonix/index.php/GPS_Data_Collection, C is a 3x3 matrix and based on this website http://perk-software.cs.queensu.ca/plus/doc/nightly/user/DeviceSonixVideo.html, C is reformed to be a 4x4 transformation matrix (referred to as T). I can do simple math to verify both are fine, but here is what I am confused:

  1. As written on both websites “% Ultrasonix uses [1yx] coordinates (first column is origin, second column is Y axis, third column is X axis)”, we can compute the missing Z axis by:
    Z axis vector = cross(X,Y). Now if [X, Y, Z] represents a rotation matrix, why norm(Y) != 1. Appreciate that norm(Y) is close to being 1, but why it is not 1.
  2. Given the rotation matrix R = [X, Y, Z] and in order to avoid the confusion left-hand and right-hand rules may bring up, I used the combined transformation matrix (R_ref = RzRyRx) as written in the bottom of this website as a reference: http://planning.cs.uiuc.edu/node102.html
    My question is
    R = R_ref or R = transpose(R_ref) ?

Any suggestion is appreciated. Thank you.

Best regards,