Problem 635. Angle between Two Vectors

Created by Richard Zapor

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The dot product relationship, a dot b = | a | | b | cos(theta), can be used to determine the acute angle between vector a and vector b ( 0 to pi ).The definition of | a | is ( a(1)a(1)+a(2)a(2)...+a(n)a(n) )^0.5.The definition of "a dot b" is a(1)b(1)+a(2)b(2)...+a(n)b(n). (wikipedia)In 3-D the angle is in the plane created by the vectors a and b.The input may be a 2-D or a 3-D vector. These represent physical models.An extension of this angular determination given vectors problem is to provide two points for each vector.
The practical application relates to Laser Trackers which best fit multiple points for lines, surfaces, annular surfaces, and other reference points.Examples:a=[1 0] (x-axis); b=[0 1] (y-axis) which intersect at 90 degrees (pi/2)theta=acos(a dot b/(|a||b|)=acos(0/(1*1))=pi/2 radiansa=[1 1 0] 45 degrees in xy plane
b=[1 1 1.414] 45 degree vector in Z above a 45 degree rotation in XY plane.theta=acos(a dot b/(|a||b|)=acos(2/(1.414*2))=pi/4 radians

The dot product relationship, a dot b = | a | | b | cos(theta), can be used to determine the acute angle between vector a and vector b ( 0 to pi ).
 
The definition of | a | is ( a(1)a(1)+a(2)a(2)...+a(n)a(n) )^0.5.

The definition of "a dot b" is a(1)b(1)+a(2)b(2)...+a(n)b(n). (wikipedia)

In 3-D the angle is in the plane created by the vectors a and b. 
 
The input may be a 2-D or a 3-D vector. These represent physical models.

An extension of this angular determination given vectors problem is to provide two points for each vector.
The practical application relates to Laser Trackers which best fit multiple points for lines, surfaces, annular surfaces, and other reference points.

Examples:

a=[1 0] (x-axis); b=[0 1] (y-axis) which intersect at 90 degrees (pi/2)

theta=acos(a dot b/(|a||b|)=acos(0/(1*1))=pi/2 radians

a=[1 1 0] 45 degrees in xy plane
b=[1 1 1.414] 45 degree vector in Z above a 45 degree rotation in XY plane.

theta=acos(a dot b/(|a||b|)=acos(2/(1.414*2))=pi/4 radians

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Last Solution submitted on Oct 11, 2021

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David Verrelli on 4 Apr 2018

Spookily similar to Problem 381 ("Angle between two vectors")....

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