Problem 635. Angle between Two Vectors

Created by Richard Zapor

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>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 ).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The definition of | a | is ( a(1)a(1)+a(2)a(2)...+a(n)a(n) )^0.5.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The definition of \"a dot b\" is a(1)b(1)+a(2)b(2)...+a(n)b(n). (wikipedia)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In 3-D the angle is in the plane created by the vectors a and b.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The input may be a 2-D or a 3-D vector. These represent physical models.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>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.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Examples:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>a=[1 0] (x-axis); b=[0 1] (y-axis) which intersect at 90 degrees (pi/2)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>theta=acos(a dot b/(|a||b|)=acos(0/(1*1))=pi/2 radians</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>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.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>theta=acos(a dot b/(|a||b|)=acos(2/(1.414*2))=pi/4 radians</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

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

Solve

Solution Stats

652 Solutions
316 Solvers

Last Solution submitted on Jan 24, 2023

Last 200 Solutions

Problem Comments

1 Comment

1 Comment

David Verrelli on 4 Apr 2018

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

Problem Recent Solvers316

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 635. Angle between Two Vectors

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers316

Suggested Problems

More from this Author260

Problem Tags

Community Treasure Hunt

Problem 635. Angle between Two Vectors

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers316

Suggested Problems

More from this Author260

Problem Tags

Community Treasure Hunt

Players