# calculation the angle betwen X-axis created due coordinate system rotation - after projection the coordinate system on the plane

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UWM on 13 Jan 2020
Answered: darova on 13 Jan 2020 I have a coordinate system XYZ (black on figure) which can be rotated around
Z axis (Zrot angle) or around Y axis (Yrot angle) or around both axisies (around Z first and then around rotated (in first rotation around Z axis) Y axis).
E.g. after rotation only around Z axis we get X'Y'Z coordinate system (see fig).
I have also the direction (point P) defined by azimuth and evation (e.g. AZ, EL) in XYZ coordinate system (blue on figure).
Rotation code are below:
% geodetic elevation and azimuth in deg(El, AZ)
Esvo = 50;
Asvo = 30;
% MatLab elevation and azimuth
Esv = Esvo;
Asv = Asvo;
if Asvo < 180; Asv=-Asvo;
else Asv = 360-Asvo;
end
r=1
dtr = pi/180;
Es=Esv*dtr % deg to rad conversion
As=Asv*dtr % deg to rad conversion
% initial position in MatLab
[x0,y0,z0] = sph2cart(As,Es,r)
poz0 = [x0 y0 z0]
% coordinate system rotation angles in deg (Zrot=Aro1; Yrot=Ero1)
Ero1 = 10;
Aro1 = 40;
% AZ change after Zrot rotation (geodetic)
Asvn = Asvo-Aro1
% chenged AZ in MatLab
if Asvn < 180; Asvn=-Asvn;
else Asvn = 360-Asvn;
end
Esv=Esv*dtr % deg to rad conversion
Asvn=Asvn*dtr % deg to rad conversion
% position after rotation around Z-axis
[x1,y1,z1] = sph2cart(Asvn,Esv,r)
poz1 = [x1 y1 z1]
% rotation around Y-axis
ROTY = [ cosd(Ero1) 0 sind(Ero1);
0 1 0;
-sind(Ero1) 0 cosd(Ero1)];
% position after two rotation (in MatLab)
poz2 = poz1*ROTY;
x2 = poz2(1);
y2 = poz2(2);
z2 = poz2(3);
I would like to create a PLANE (red on figure) which is perpendicular to direction P (direction P will be normal to PLANE) and then calculate angle B
which is the difference betwen X and X' axisies orthogonally projected on PLANE (red on fig).
Any suggestions how can be it done?

darova on 13 Jan 2020
Here is an idea of how to project axis onto plane:
• Unit vector (normal) of a plane: • Distance to plane: • Axis projected onto plane:  Having both projected axes: 