Calculate the 3D angle between two vectors

How do you calculate the angles between two vectors in order to generate a direction cosine matrix? I have MATLAB, Simulink, and Aerospace Toolkit/Toolbox. For reference, I am trying to use MATLAB to calculate a rotation matrix between the body frame of an object and (preferably) a fixed-frame or (not so preferably) an inertial-frame.

Answer 1

[Roger Stafford]

The angle between two three-element vectors, P1 and P2, can be calculated using matlab in the following way:

a = atan2(norm(cross(P1,P2)),dot(P1,P2)); % Angle in radians

The angle will lie between 0 and pi radians. To get degrees use ‘atan2d’.

Note: However, the cosine of such an angle can be calculated as:

cosine of the angle = dot(P1,P2)/(norm(P1)*norm(P2))

No need to compute the angle itself.

[Paul Huter]

I am looking for the angles between each component of the three components of the vectors.

[Roger Stafford]

Let a vector be V = [x;y;z]. The three direction cosines would be: V/norm(V). For the three corresponding angles, either use the ‘acos’ function on each such cosine value or else do the somewhat more accurate:

ax = atan2(norm(cross(V,[1;0;0])),dot(V,[1;0;0]));
ay = atan2(norm(cross(V,[0;1;0])),dot(V,[0;1;0]));
az = atan2(norm(cross(V,[0;0;1])),dot(V,[0;0;1]));

Note: These latter are the same as writing:

ax = atan2(sqrt(y^2+z^2),x);
ay = atan2(sqrt(z^2+x^2),y);
az = atan2(sqrt(x^2+y^2),z);

[Zhafirah Ariana]

I tried this and it worked! But when i use this for array it didn't work. Do you have any idea why this happened?

[Bruno Luong]

For array of size 3 x N where 3 rows are x-y-z coordinates you might replace the

• CROSS with CROSS_DIM1
• DOT with DOT_DIM1
• NORM(X) with vecnorm(x, 2, 1) or sqrt(sum(x.^2,1)).
• "^" with ".^"

CROSS_DIM1

function c = cross_dim1(a,b)
% c = cross_dim1(a,b)
% Calculate cross product along the first dimension
% NOTE: auto expansion allowed

c = zeros(max(size(a),size(b)));
c(1,:) = a(2,:).*b(3,:)-a(3,:).*b(2,:);
c(2,:) = a(3,:).*b(1,:)-a(1,:).*b(3,:);
c(3,:) = a(1,:).*b(2,:)-a(2,:).*b(1,:);

end % cross_dim1

and DOT_DIM1

function d = dot_dim1(a,b)
% d = dot_dim1(a,b)
% Calculate dot product along the first dimension
% NOTE: auto expansion allowed

d = sum(a.*b,1);

end % dot_dim1

Answer 2

[Image Analyst]

See https://www.mathworks.com/matlabcentral/answers/318526-direction-cosine-matrix-using-left-handed-coordinate-system

[Paul Huter]

I do not have the angle, just the two vectors.

Answer 3

[Bingo]

function r = vrrotvec(a, b, options)

%VRROTVEC Calculate a rotation between two vectors.

% R = VRROTVEC(A, B) calculates a rotation needed to transform

% a 3d vector A to a 3d vector B.

[Image Analyst]

You forgot to provide the body of the function.

[Matlab Helmstaedter]

Its an inbuilt function.

[John D'Errico]

Not really. It is only there IF you just happen to have the correct toolbox.

help vrrotvec
'vrrotvec' requires Simulink 3D Animation.

Otherwise, you need to use the alteriatives.

[Image Analyst]

I don't have that toolbox. Simulink 3D Animation is not on the Products list on the right. I will add it.