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MathWorks Support Team on 22 Jun 2011
Commented: Bruno Luong on 3 Dec 2022
Accepted Answer: MathWorks Support Team
How can I determine the angle between two vectors in MATLAB?
I have two vectors. Is there a MATLAB function that can determine the angle between them?
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MathWorks Support Team on 27 May 2020
Edited: MathWorks Support Team on 27 May 2020
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There is no in-built MATLAB function to find the angle between two vectors. As a workaround, you can try the following:
CosTheta = max(min(dot(u,v)/(norm(u)*norm(v)),1),-1);
ThetaInDegrees = real(acosd(CosTheta));
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Roger Stafford on 9 Jul 2015
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https://www.mathworks.com/matlabcentral/answers/101590-how-can-i-determine-the-angle-between-two-vectors-in-matlab#comment_297320
Edited: James Tursa on 9 Jul 2015
@Mathworks Support Team: I think you will find that Mathworks' 'atan2' function is more accurate than 'acos' for angles that are near zero or pi radians. Just look at a plot of acos(x) to see why.
James Tursa on 24 May 2018
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@Felix: Did you expect something different?
Johannes Kalliauer on 14 Jan 2020
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@MathWorks Support Team
u = [1 2 0];
v = [1 0 0];
CosTheta = dot(u,v)/(norm(u)*norm(v));
ThetaInDegrees = acosd(CosTheta);
Almost linear Vectors (Angle of say 8E-10rad) might leed to problems since CosTheta might be slightly bigger than 1, due to nummerical double precission (say something around 1+2E-16) and leed to imaginary angles
u = [1 2 0];
v = [1 0 0];
CosTheta = max(min(dot(u,v)/(norm(u)*norm(v)),1,-1);
ThetaInDegrees = acosd(CosTheta);
Johannes Kalliauer on 3 Feb 2020
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@MathWorks Support Team
u=[0.272379472472602111022302462516 1.08301805439555555910790149937 -0.359366773005409555910790149937];
v=[0.2898030626583580555111512312578 1.15229663744866689137553104956 -0.382354774507524222044604925031];
CosTheta = (dot(u,v) / (norm(u)*norm(v)));
if abs(CosTheta)>1
error('MATLAB:odearguments:NumericPrecision','Matlab has numerical issues in calculated angle')
end
leads to an error (imaginär angle), since CosTheta=1+2.22044604925031e-16>1
Solution would be
CosTheta = max(min(dot(u,v)/(norm(u)*norm(v)),1,-1);
ThetaInDegrees = real(acosd(CosTheta));
.
Akihumi on 27 May 2020
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Hi, did you miss out a bracket for the min? I got an error and only resolve it with the following code instead.
CosTheta = max(min(dot(u,v)/(norm(u)*norm(v)),1),-1);
ThetaInDegrees = real(acosd(CosTheta));
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More Answers (2)
Pierre-Pascal on 11 Jan 2016
So why doesn't matlab give us a function for that instead of having us look endlessly on forums?
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James Tursa on 9 Jul 2015
Edited: James Tursa on 5 Jan 2019
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This topic has been discussed many times on the Newsgroup forum ... if I looked hard enough I'm sure I could find several Roger Stafford posts from many years ago on this. E.g., here is one of them:
https://www.mathworks.com/matlabcentral/answers/161515-how-do-i-find-the-angle-between-a-vector-and-a-line
The basic acos formula is known to be inaccurate for small angles. A more robust method is to use both the sin and cos of the angle via the cross and dot functions. E.g.,
atan2(norm(cross(u,v)),dot(u,v));
An extreme case to clearly show the difference:
>> a = 1e-10 % start with a very small angle
a =
1e-10
>> u = 4*[1 0 0] % arbitrary non-unit vector in X direction
u =
4 0 0
>> v = 5*[cos(a) sin(a) 0] % vector different from u by small angle
v =
5 5e-10 0
>> acos(dot(u,v)/(norm(u)*norm(v))) % acos formulation does not recover the small angle
ans =
>> atan2(norm(cross(u,v)),dot(u,v)) % atan2 formulation does recover the small angle
ans =
1e-10
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Johannes Kalliauer on 14 Jan 2020
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Thanks, sometimes even imaginäry Angles occur if using acos-function.
James Tursa on 3 Feb 2020
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To get a full circle result where "direction" of the angle is important, see this link for one possible strategy:
https://www.mathworks.com/matlabcentral/answers/501449-angle-betwen-two-3d-vectors-in-the-range-0-360-degree
Bruno Luong on 3 Dec 2022
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@Felix Fischer If you want to find angles of multiple vector pairs put in matrix, use vecnorm rather than norm.
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