Angles between two n-dimensional vectors in Python

Question

I need to determine the angle s  between two n-dimensional vectors in Python  For example  the input can be two lists like the following   1 2 3 4  and  6 7 8 9

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Note  all of the other answers here will fail if the two vectors have either the same direction  ex   1  0  0    1  0  0   or opposite directions  ex   -1  0  0    1  0  0     Here is a function which will correctly handle these cases   import numpy as np  def unit vector vector           Returns the unit vector of the vector           return vector   np linalg norm vector   def angle between v1  v2           Returns the angle in radians between vectors  v1  and  v2                  gt  gt  gt  angle between  1  0  0    0  1  0               1 5707963267948966              gt  gt  gt  angle between  1  0  0    1  0  0               0 0              gt  gt  gt  angle between  1  0  0    -1  0  0               3 141592653589793             v1 u   unit vector v1      v2 u   unit vector v2      return np arccos np clip np dot v1 u  v2 u   -1 0  1 0

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If you re working with 3D vectors  you can do this concisely using the toolbelt vg  It s a light layer on top of numpy   import numpy as np import vg  vec1   np array  1  2  3   vec2   np array  7  8  9    vg angle vec1  vec2    You can also specify a viewing angle to compute the angle via projection   vg angle vec1  vec2  look vg basis z    Or compute the signed angle via projection   vg signed angle vec1  vec2  look vg basis z    I created the library at my last startup  where it was motivated by uses like this  simple ideas which are verbose or opaque in NumPy

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Easy way to find angle between two vectors works for n-dimensional vector    Python code   import numpy as np  vector1    1 0 0  vector2    0 1 0   unit vector1   vector1   np linalg norm vector1  unit vector2   vector2   np linalg norm vector2   dot product   np dot unit vector1  unit vector2   angle   np arccos dot product   angle in radian

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Using numpy and taking care of BandGap s rounding errors   from numpy linalg import norm from numpy import dot import math  def angle between a b     arccosInput   dot a b  norm a  norm b    arccosInput   1 0 if arccosInput  gt  1 0 else arccosInput   arccosInput   -1 0 if arccosInput  lt  -1 0 else arccosInput   return math acos arccosInput    Note  this function will throw an exception if one of the vectors has zero magnitude  divide by 0

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For the few who may have  due to SEO complications  ended here trying to calculate the angle between two lines in python  as in  x0  y0    x1  y1  geometrical lines  there is the below minimal solution  uses the shapely module  but can be easily modified not to    from shapely geometry import LineString import numpy as np  ninety degrees rad   90 0   np pi   180 0  def angle between line1  line2       coords 1   line1 coords     coords 2   line2 coords      line1 vertical    coords 1 1  0  - coords 1 0  0      0 0     line2 vertical    coords 2 1  0  - coords 2 0  0      0 0        Vertical lines have undefined slope  but we know their angle in rads is   90     p 180     if line1 vertical and line2 vertical            Perpendicular vertical lines         return 0 0     if line1 vertical or line2 vertical            90   - angle of non-vertical line         non vertical line   line2 if line1 vertical else line1         return abs  90 0   np pi   180 0  - np arctan slope non vertical line         m1   slope line1      m2   slope line2       return np arctan  m1 - m2   1   m1 m2    def slope line         Assignments made purely for readability  One could opt to just one-line return them     x0   line coords 0  0      y0   line coords 0  1      x1   line coords 1  0      y1   line coords 1  1      return  y1 - y0     x1 - x0    And the use would be   gt  gt  gt  line1   LineString   0  0    0  1      vertical  gt  gt  gt  line2   LineString   0  0    1  0      horizontal  gt  gt  gt  angle between line1  line2  1 5707963267948966  gt  gt  gt  np degrees angle between line1  line2   90 0

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The other possibility is using just numpy and it gives you the interior angle  import numpy as np  p0    3 5  6 7  p1    7 9  8 4  p2    10 8  4 8        compute angle  in degrees  for p0p1p2 corner Inputs      p0 p1 p2 - points in the form of  x y       v0   np array p0  - np array p1  v1   np array p2  - np array p1   angle   np math atan2 np linalg det  v0 v1   np dot v0 v1   print np degrees angle    and here is the output   In  2   p0  p1  p2    3 5  6 7    7 9  8 4    10 8  4 8   In  3   v0   np array p0  - np array p1   In  4   v1   np array p2  - np array p1   In  5   v0 Out 5   array  -4 4  -1 7    In  6   v1 Out 6   array   2 9  -3 6    In  7   angle   np math atan2 np linalg det  v0 v1   np dot v0 v1    In  8   angle Out 8   1 8802197318858924  In  9   np degrees angle  Out 9   107 72865519428085

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David Wolever s solution is good  but  If you want to have signed angles you have to determine if a given pair is right or left handed  see wiki for further info    My solution for this is   def unit vector vector           Returns the unit vector of the vector        return vector   np linalg norm vector   def angle vector1  vector2           Returns the angle in radians between given vectors        v1 u   unit vector vector1      v2 u   unit vector vector2      minor   np linalg det          np stack  v1 u -2    v2 u -2               if minor    0          raise NotImplementedError  Too odd vectors          return np sign minor    np arccos np clip np dot v1 u  v2 u   -1 0  1 0     It s not perfect because of this NotImplementedError but for my case it works well  This behaviour could be fixed  cause handness is determined for any given pair  but it takes more code that I want and have to write

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Building on sgt pepper s great answer and adding support for aligned vectors plus adding a speedup of over 2x using Numba   njit cache True  nogil True  def angle vector1  vector2           Returns the angle in radians between given vectors        v1 u   unit vector vector1      v2 u   unit vector vector2      minor   np linalg det          np stack  v1 u -2    v2 u -2               if minor    0          sign   1     else          sign   -np sign minor      dot p   np dot v1 u  v2 u      dot p   min max dot p  -1 0   1 0      return sign   np arccos dot p    njit cache True  nogil True  def unit vector vector           Returns the unit vector of the vector           return vector   np linalg norm vector   def test angle        def npf x           return np array x  dtype float      assert np isclose angle npf  1  1    npf  1   0      pi   4      assert np isclose angle npf  1  0    npf  1   1     -pi   4      assert np isclose angle npf  0  1    npf  1   0      pi   2      assert np isclose angle npf  1  0    npf  0   1     -pi   2      assert np isclose angle npf  1  0    npf  1   0      0      assert np isclose angle npf  1  0    npf  -1  0      pi      timeit results without Numba   359   s    2 86   s per loop  mean    std  dev  of 7 runs  1000 loops each    And with   151   s    820 ns per loop  mean    std  dev  of 7 runs  10000 loops each

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Using numpy  highly recommended   you would do   from numpy import  array  dot  arccos  clip  from numpy linalg import norm  u   array  1  2 3 4   v       c   dot u v  norm u  norm v    - gt  cosine of the angle angle   arccos clip c  -1  1     if you really want the angle

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import math  def dotproduct v1  v2     return sum  a b  for a  b in zip v1  v2    def length v     return math sqrt dotproduct v  v    def angle v1  v2     return math acos dotproduct v1  v2     length v1    length v2      Note  this will fail when the vectors have either the same or the opposite direction  The correct implementation is here  https   stackoverflow com a 13849249 71522

[python] Angles between two n-dimensional vectors in Python

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