Setting a 3rd vector with specific angles to 2 other vectors

Hello, I want to define 2 vectors in the 3D coordinate space (A and B) and the angle between them should be 92.1 degrees. I then want to set a 3rd vector (C) which will be at a 84.4 degrees angle with A and 86.2 degrees with B. Here is the code I am using for that:

% Define angles I want 
A_B = 92.1;
A_C = 84.4;
B_C = 86.2;

% Create a unit vector A
A = [0; 0; 1];

% Define B by rotating A by A_B angle around x-axis
rotationx = [1        0            0     ; ... 
             0    cosd(A_B)   -sind(A_B) ; ...
             0    sind(A_B)   cosd(A_B)] ;

B = rotationx*A;

% Define C with respect to A by rotating A by A_C angle around y-axis
rotationy = [cosd(A_C)        0         sind(A_C) ; ... 
                 0            1             0     ; ...
             -sind(A_C)       0         cosd(A_C)];

C = rotationy*A;

% Calculate the angle difference between current C and B
angleCurrent = atan2d(norm(cross(C,B)), dot(C,B));

% Find the difference between the current angle and the C-B value I want to
% get
angleDiff = B_C-angleCurrent;

% Move C again around the z-axis
rotationz = [cosd(angleDiff)    -sind(angleDiff)      0; ... 
             sind(angleDiff)    cosd(angleDiff)      0; ...
                   0                  0              1];

C = rotationz*C;

With this code I get the following values. Looks like the angle between A_B and A_C is fine, but the B_C is very slightly off. I guess I should move my C vector around the y and z axes at the same time instead of moving it in separate steps with repsect to A and B. Am I correct? If so, how should the rotation matrix look? Thanks!

A_B = 92.1;
A_C = 84.4;
B_C = 86.2223;

Answer 1

[David Goodmanson]

Hi Huseyin,

The method you are using gets you into an iterative process of successive rotations about the x and z axis that eventually converge on the result. The method below for C uses a standard rotation about the y axis like you started with, followed by a rotation about z by a yet to be determined angle theta. Then you use a bit of algebral to find theta.

A_B = 92.1;
A_C = 84.4;
B_C = 86.2;
A = [0;                 0;                 1        ];
% find B, per your rotation
B = [0;                -sind(A_B);         cosd(A_B)];

% rotate in spherical coordinates to find C.
% note that that angle (A_C) is still satisfied
% C = [sind(A_C)*cosd(theta);   sind(A_C)*sind(theta);   cosd(A_C)];
% now  BdotC = -sind(A_B)*sind(A_C)*sind(theta) + cosd(A_B)*cosd(A_C) = cosd(B_C)
% solve
sindtheta = (cosd(A_B)*cosd(A_C)-cosd(B_C))/(sind(A_B)*sind(A_C));
theta = asind(sindtheta)
Cnew = [sind(A_C)*cosd(theta); sind(A_C)*sind(theta); cosd(A_C)]

% check, should be small
dot(A,B)-cosd(A_B)
dot(A,Cnew)-cosd(A_C)
dot(B,Cnew)-cosd(B_C)

D = C -Cnew

    theta =  -4.0273
    Cnew =
         0.9928
        -0.0699
         0.0976
    D =         
         1.0e-03 *
         0.0273
         0.3886
         0

Answer 2

[Ozzy]

Awesome, thanks for the answer. Just so I understood this correctly, after the initial y-axis rotation to find C what we are doing here essentially is measuring the angle between the vector C and vectors A&B iteratively and rotating C between them until we minimize some sort of an error, is that correct? If so, is this an error caused by the rotation matrix?

[David Goodmanson]

Huseyin,

I mentioned that what you were doing was the start of an iterative process. The method I used is not iterative. Starting with A, it defines C by using a known rotation angle (A_C) about y, followed by an unknown rotation angle theta about z. It finds the dot product of B and C, which has to equal cosd(B_C), and solves for sind(theta) algebraically in one step. Then it uses that value of theta to find C.