vector projection on rotated coordinate system

I have to vectors: red (XR, YR, ZR) and blue (XB, YB, ZB) in XYZ coordinate

system. I would like to calculate projection of these two vectors on X'Y', X'Z' and Y'Z' planes

where X'Y'Z' is rotaded, along Z (angle 'alpha') and Y (angle 'beta') axises, coordinate system XYZ.

Vectors stay in old XYZ system (are not rotated). Only system is rotaded along two axisies.

Any suggestions how this could be done?

Answer 1

[Matt J]

I'll assume your given vectors are in a 3x2 matrix called columnVectors, that your angles are in degrees, and that rotations are done in Z first and Y second.

Rz=@(x) [cosd(x),-sind(x),0 ; ...
        sind(x),cosd(x),0 ;...
        0 0 1];
    
Ry = @(x) [cosd(x), 0, sind(x); 0 1 0; -sind(x), 0, cosd(x) ];

R=Ry(beta)*Rz(alpha); %ANGLES ARE IN DEGREES!!



[XpYp,XpZp,YpZp]=deal(R.'*columnVectors);

XpYp(3,:)=0; %X'Y'
XpZp(2,:)=0; %X'Z'
YpZp(1,:)=0; %Y'Z'

XpYp=R*XpYp;
XpZp=R*XpZp;
YpZp=R*YpZp;

[UWM]

It works. Thank you.