How do I calculate the normal vector of a line segment

Question

Suppose I have a line segment going from  x1 y1  to  x2 y2   How do I calculate the normal vector perpendicular to the line   I can find lots of stuff about doing this for planes in 3D  but no 2D stuff   Please go easy on the maths  links to worked examples  diagrams or algorithms are welcome   I m a programmer more than I m a mathematician

User · Accepted Answer

If we define dx   x2 - x1 and dy   y2 - y1  then the normals are  -dy  dx  and  dy  -dx   Note that no division is required  and so you re not risking dividing by zero

User · Answer

Another way to think of it is to calculate the unit vector for a given direction and then apply a 90 degree counterclockwise rotation to get the normal vector   The matrix representation of the general 2D transformation looks like this   x    x cos t  - y sin t  y    x sin t    y cos t    where  x y  are the components of the original vector and  x   y   are the transformed components   If t   90 degrees  then cos 90    0 and sin 90    1   Substituting and multiplying it out gives   x    -y y     x   Same result as given earlier  but with a little more explanation as to where it comes from

User · Answer

m1    y2 - y1     x2 - x1    if perpendicular two lines   m1 m2   -1   then   m2   -1   m1   if  m1    0  then your line should have an equation like x   b   y   m2 x   b   b is offset of new perpendicular line      b is something if you want to pass it from a point you defined

User · Answer

This question has been posted long time ago  but I found an alternative way to answer it  So I decided to share it here  Firstly  one must know that  if two vectors are perpendicular  their dot product equals zero  The normal vector  x  y   is perpendicular to the line connecting  x1 y1  and  x2 y2   This line has direction  x2-x1 y2-y1   or  dx dy   So    x  y    dx dy    0 x  dx   y  dy   0   The are plenty of pairs  x  y   that satisfy the above equation  But the best pair that ALWAYS satisfies is either  dy -dx  or  -dy dx

[math] How do I calculate the normal vector of a line segment?

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