Using atan2 to find angle between two vectors

Question

I understand that   atan2 vector y  vector x    the angle between the vector and the X axis   But I wanted to know how to get the angle between two vectors using atan2  So I came across this solution   atan2 vector1 y - vector2 y  vector1 x - vector2 x    My question is very simple   Will the two following formulas produce the same number    atan2 vector1 y - vector2 y  vector1 x - vector2 x  atan2 vector2 y - vector1 y  vector2 x - vector1 x    If not  How do I know what vector comes first in the subtractions

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The formula  angle vector b vector a   that I sent  give results   in the four quadrants and for any coordinates xa ya and xb yb   For coordinates xa ya 0 and or xb yb 0 is undefined   The angle can be bigger or smaller than pi  and can be positive  or negative

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atan2 vector1 y - vector2 y  vector1 x - vector2 x    is the angle between the difference vector  connecting vector2 and vector1  and the x-axis  which is problably not what you meant   The  directed  angle from vector1 to vector2 can be computed as  angle   atan2 vector2 y  vector2 x  - atan2 vector1 y  vector1 x     and you may want to normalize it to the range  0  2 p    if  angle  lt  0    angle    2   M PI      or to the range  -p  p    if  angle  gt  M PI           angle -  2   M PI    else if  angle  lt   -M PI    angle    2   M PI

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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      xb yb and xa ya are the coordinates of the two vectors

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I think a better formula was posted here  http   www mathworks com matlabcentral answers 16243-angle-between-two-vectors-in-3d  angle   atan2 norm cross a b    dot a b     So this formula works in 2 or 3 dimensions  For 2 dimensions this formula simplifies to the one stated above

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If you care about accuracy for small angles  you want to use this   angle   2 atan2      b  a -   a  b          b  a     a  b      Where      means absolute value  AKA  length of the vector   See https   math stackexchange com questions 1143354 numerically-stable-method-for-angle-between-3d-vectors 1782769  However  that has the downside that in two dimensions  it loses the sign of the angle

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Nobody pointed out that if you have a single vector  and want to find the angle of the vector from the X axis  you can take advantage of the fact that the argument to atan2   is actually the slope of the line  or  delta Y   delta X   So if you know the slope  you can do the following   given   A   angle of the vector line you wish to determine  from the X axis    m   signed slope of the vector line   then   A   atan2 m  1   Very useful

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Here a little program in Python that uses the angle between vectors to determine if a point is inside or outside a certain polygon  import sys import numpy as np import matplotlib pyplot as plt import matplotlib patches as patches from shapely geometry import Point  Polygon from pprint import pprint    Plot variables x min  x max   -6  12 y min  y max   -3  8 tick interval   1 FIG SIZE    10  10  DELTA ERROR   0 00001 IN BOX COLOR    yellow  OUT BOX COLOR    black    def angle between v1  v2           Returns the angle in radians between vectors  v1  and  v2          The sign of the angle is dependent on the order of v1 and v2         so acos norm dot v1  v2    does not work and atan2 has to be used  see          https   stackoverflow com questions 21483999 using-atan2-to-find-angle-between-two-vectors             arg1   np cross v1  v2      arg2   np dot v1  v2      angle   np arctan2 arg1  arg2      return angle   def point inside point  border           Returns True if point is inside border polygon and False if not         Arguments           point  x  y in shapely geometry Point type          border   x1 y1  x2 y2        xn yn  in shapely geomettry Polygon type                 assert len border exterior coords   gt  2            number of points in the polygon must be  gt  2       point   np array point      side1   np array border exterior coords 0   - point     sum angles   0     for border point in border exterior coords 1            side2   np array border point  - point         angle   angle between side1  side2          sum angles    angle         side1   side2        if wn is 1 then the point is inside     wn   sum angles   2   np pi     if abs wn - 1   lt  DELTA ERROR          return True     else          return False   class MainMap          classmethod     def settings cls  fig size             set the plot outline  including axes going through the origin         cls fig  cls ax   plt subplots figsize fig size          cls ax set xlim -x min  x max          cls ax set ylim -y min  y max          cls ax set aspect 1          tick range x   np arange round x min    10  x max - x min    tick interval  10  1               x max   0 1  step tick interval          tick range y   np arange round y min    10  y max - y min    tick interval  10  1                y max   0 1  step tick interval          cls ax set xticks tick range x          cls ax set yticks tick range y          cls ax tick params axis  both   which  major   labelsize 6          cls ax spines  left   set position  zero           cls ax spines  right   set color  none           cls ax spines  bottom   set position  zero           cls ax spines  top   set color  none         classmethod     def get ax cls           return cls ax       staticmethod     def plot            plt tight layout           plt show     class PlotPointandRectangle MainMap        def   init   self  start point  rectangle polygon  tolerance 0            self current object   None         self currently dragging   False         self fig canvas mpl connect  key press event   self on key          self plot types     o    o-           self plot type   1         self rectangle   rectangle polygon            define a point that can be moved around         self point   patches Circle  start point x  start point y   0 10              alpha 1          if point inside start point  self rectangle                color   IN BOX COLOR         else               color   OUT BOX COLOR         self point set color  color          self ax add patch self point          self point set picker tolerance          cv point   self point figure canvas         cv point mpl connect  button release event   self on release          cv point mpl connect  pick event   self on pick          cv point mpl connect  motion notify event   self on motion           self plot rectangle        def plot rectangle self           x    point 0  for point in self rectangle exterior coords          y    point 1  for point in self rectangle exterior coords            y   self rectangle y         self rectangle plot    self ax plot x  y              self plot types self plot type   color  r   lw 0 4  markersize 2       def on release self  event           self current object   None         self currently dragging   False      def on pick self  event           self currently dragging   True         self current object   event artist      def on motion self  event           if not self currently dragging              return         if self current object    None              return          point   Point event xdata  event ydata          self current object center   point x  point y         if point inside point  self rectangle                color   IN BOX COLOR         else               color   OUT BOX COLOR         self current object set color  color           self point figure canvas draw        def remove rectangle from plot self           try              self rectangle plot remove           except ValueError              pass      def on key self  event             with  space  toggle between just points or points connected with           lines         if event key                     self plot type    self plot type   1    2             self remove rectangle from plot               self plot rectangle               self point figure canvas draw     def main start point  rectangle        MainMap settings FIG SIZE      plt me   PlotPointandRectangle start point  rectangle    pylint  disable unused-variable     MainMap plot    if   name         main         try          start point   Point  float val  for val in sys argv 1  split         except IndexError          start point  Point 0  0       border points     -2  -2                         1  1                         3  -1                         3  3 5                         4  1                         5  1                         4  3 5                         5  6                         3  4                         3  5                         -0 5  1                         -3  1                         -1  -0 5                                             border points polygon   Polygon border points      main start point  border points polygon

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The proper way to do it is by find the sine of the angle using the cross product  and the cosine of the angle using the dot product and combine the two with the Atan2   function   In C  this is   public struct Vector2       public double X  Y            lt summary gt          Returns the angle between two vectos          lt  summary gt      public static double GetAngle Vector2 A  Vector2 B                    A  B     A   B  COS                A  B     A   B  SIN             return Math Atan2 Cross A B   Dot A B               public double Magnitude   get   return Math Sqrt Dot this this             public static double Dot Vector2 A  Vector2 B                return A X B X A Y B Y            public static double Cross Vector2 A  Vector2 B                return A X B Y-A Y B X          class Program       static void Main string   args                Vector2 A new Vector2     X 5 45  Y 1 12           Vector2 B new Vector2     X -3 86  Y 4 32             double angle Vector2 GetAngle A  B    180 Math PI             angle   120 16850967865749           See test case above in GeoGebra

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As a complement to the answer of  martin-r one should note that it is possible to use the sum difference formula for arcus tangens   angle   atan2 vec2 y  vec2 x  - atan2 vec1 y  vec1 x   angle   -atan2 vec1 x   vec2 y - vec1 y   vec2 x  dot vec1  vec2           where dot   vec1 x   vec2 x    vec1 y   vec2 y    Caveat 1  make sure the angle remains within -pi      pi Caveat 2  beware when the vectors are getting very similar  you might get extinction in the first argument  leading to numerical inaccuracies

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You don t have to use atan2 to calculate the angle between two vectors  If you just want the quickest way  you can use dot v1  v2   v1   v2  cos A to get  A   Math acos  dot v1  v2   v1 length   v2 length

[math] Using atan2 to find angle between two vectors

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