# How can I determine the angle between two vectors in MATLAB?

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MathWorks Support Team
on 22 Jun 2011

Commented: DGM
on 4 Jul 2022

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?

### Accepted Answer

MathWorks Support Team
on 27 May 2020

Edited: MathWorks Support Team
on 27 May 2020

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));

##### 10 Comments

Dan Hendrickson
on 27 May 2020

Akihumi,

Thanks for pointing that out, I have updated the MathWorks answer.

Best,

Dan

### More Answers (8)

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?

##### 1 Comment

William Chamberlain
on 27 Jul 2016

James Tursa
on 9 Jul 2015

Edited: James Tursa
on 5 Jan 2019

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:

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 =

0

>> atan2(norm(cross(u,v)),dot(u,v)) % atan2 formulation does recover the small angle

ans =

1e-10

##### 4 Comments

James Tursa
on 3 Feb 2020

Gabor Bekes
on 15 Sep 2016

Edited: Gabor Bekes
on 15 Sep 2016

This does the same thing, also capable of determining the angle of higher (than one) dimensional subspaces.

subspace(vector1,vector2)

##### 1 Comment

Aras
on 3 May 2018

Edited: Aras
on 3 May 2018

This method needs to be used carefully because it provides an angle between 0 and π/2 radians, instead of between 0 and π.

E.g., the angle between vectors [1, 0] and [-1, 0] are given as 0, while the result is expected to be π, considering their opposite directions.

Daniel Vasilaky
on 9 Jul 2015

Edited: Walter Roberson
on 15 Sep 2015

acosd(CosTheta)

will give you the same answer.

##### 0 Comments

Boris Povazay
on 17 Jun 2018

Just a note on how to vectorize the whole thing: (semicolons purposely omitted to see the intermediate results)

u = [1 2 0];

v = [1 0 0];

C=cross(u,v)

NC=norm(C)

D=dot(u,v)

ThetaInDegrees = atan2d(NC,D)

Rep=5

uf = repmat(u,5,1)

vf = repmat(v,5,1)

vC=cross(uf,vf,2) %vectorized

vNC=vecnorm(vC,2,2) % since only z-rotation is allowed anyway, this is equivalent to: vNC=vC(:,3)

vD=dot(uf,vf,2)

vThetaInDegrees = mean(atan2d(vNC,vD))

or in short (the hard to read variant)

VThetaInDegrees =atan2d( vecnorm(cross(Vu,Vv,2),2,2) , dot(Vu,Vv,2) )

##### 2 Comments

Jan
on 17 Jun 2018

theodore panagos
on 29 Oct 2018

Coordinates of two vectors xb,yb and xa,ya .

angle(vector.b,vector.a)=pi/2*((1+sgn(xa))*(1-sgn(ya^2))-(1+sgn(xb))*(1-sgn(yb^2)))

+pi/4*((2+sgn(xa))*sgn(ya)-(2+sgn(xb))*sgn(yb))

+sgn(xa*ya)*atan((abs(xa)-abs(ya))/(abs(xa)+abs(ya)))

-sgn(xb*yb)*atan((abs(xb)-abs(yb))/(abs(xb)+abs(yb)))

##### 0 Comments

Mahaveer Singh
on 2 May 2021

Edited: Mahaveer Singh
on 4 May 2021

function angle_in_degrees = vector2angle(u,v)

a= sqrt(u(1)^2+u(2)^2+u(3)^2);

b=sqrt(v(1)^2+v(2)^2+v(3)^2);

c=0;

for i=1:1:numel(u)

c=c+u(i)*v(i);

end

angle_in_degrees=acos(c/(a*b))*180/pi

end

##### 3 Comments

Mahaveer Singh
on 4 May 2021

%other one is

function angle_in_degrees = vector2angle(u,v)

a= sqrt(u(1)^2+u(2)^2+u(3)^2);

b=sqrt(v(1)^2+v(2)^2+v(3)^2);

angle_in_degrees=acos(dot(u,v)/(a*b))*180/pi

end

Dhritishman
on 3 Jul 2022

Currently, there is no built-in MATLAB function to calculate the angle between two vectors. However, you can use dot product property of two vectors to find the angle:

cosOfAngle = max(min(dot(u,v)/(norm(u)*norm(v)),1),-1);

angleInDegrees = real(acosd(cosOfAngle));

##### 1 Comment

DGM
on 4 Jul 2022

This is essentially a duplicate of the main answer with the variable names slightly changed.

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