# Rotate Basis Vectors Programmatically

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Mohammad MSBR on 30 Sep 2022
Commented: Mohammad MSBR on 2 Oct 2022
I have six 6-dimensional basis vectors, i.e., that are orthogonal. I wonder how I can rotate these 6 vectors programatically in the 6D space to build new basis vectors. In other words, is there a way to parameterize these basis vectors so that I can change them without losing orthogonality?
A = [a1,a2,...,a6];
B = [b1,b2,...,b6];
C = [c1,c2,...,c6];
D = [d1,d2,...,d6];
E = [e1,e2,...,e6];
F = [f1,f2,...,f6];
Thank you.

Jan on 30 Sep 2022
See: FEX: Rotation Matrix This creates N-dimensional rotation matrices.
Mohammad MSBR on 1 Oct 2022
Thank you, Jan.
The rotation matrix constructed in your suggested code works well. many thanks.

Chunru on 30 Sep 2022
Edited: Chunru on 30 Sep 2022
V1 = orth(randn(6)) % your original orthonormal basis
V1 = 6×6
-0.4706 0.7349 -0.0145 0.2908 -0.2682 0.2859 -0.3058 -0.0469 0.1456 0.2725 -0.2311 -0.8692 -0.0524 -0.3243 0.7312 0.0009 -0.5190 0.2967 -0.2590 0.3046 0.5051 -0.5918 0.4612 -0.1489 -0.7004 -0.4887 -0.0798 0.2369 0.3948 0.2287 -0.3532 -0.1447 -0.4272 -0.6594 -0.4865 -0.0169
% Then you can apply any other orthonormal basis to it
% For example,
V2 = orth(randn(6)); % get another orthonormal basis
Vnew = V2*V1; % this is the transform of the original orthonormal basis
Vnew*Vnew' % to demonstrate the oorthonormal property
ans = 6×6
1.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 1.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 1.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 1.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 1.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 1.0000
% If you want to control the rotation with angle in N-D space
% Rotate on hyperplane i-j by theta
i=2; j=4; % for example
theta = 5; % deg
R = eye(6); % 6D
R([i j], [i j]) = [cosd(theta) -sind(theta); sind(theta) cosd(theta)]
R = 6×6
1.0000 0 0 0 0 0 0 0.9962 0 -0.0872 0 0 0 0 1.0000 0 0 0 0 0.0872 0 0.9962 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000
% Then you can have a series of rotation matrices and you can put them
% together as one rotation matrices
Mohammad MSBR on 2 Oct 2022
Correct.

Mohammad MSBR on 30 Sep 2022
Thank you both for very helpful information.
The only missing point is the order of rotations. For a 6D space, I have 6 rotattions. Is there a way to pick a sequence without losing generality?
In other words, can I lead to the same beses by starting with rotations in different hyperplanes, or does starting with a specific hyperplane always lead to a unique configuration?
Thank you so much.
Mohammad MSBR on 2 Oct 2022
Thank you.

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