Lesson HOW TO find the angle between two vectors in a coordinate plane

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HOW TO find the angle between two vectors in a coordinate plane


Let assume that two vectors  u  and  v  are given in a coordinate plane via their components  u = (a,b)  and  v = (c,d).
How to find the angle between these vectors?


    The cosines of the angle  alpha  between the vectors  u = (a,b)  and  v = (c,d)  in a coordinate plane is equal to  %28a%2Ac+%2B+b%2Ad%29%2F%28sqrt%28a%5E2%2Bb%5E2%29%2Asqrt%28c%5E2%2Bd%5E2%29%29.    


Example 1
Find the angle  alpha  between the vectors  u = (2,3)  and  v = (4,5)  in a coordinate plane.

Solution
The cosines of the angle  alpha  between the vectors  u = (2,3)  and  v = (4,5)  is equal to   cos%28alpha%29 = %282%2A4+%2B+3%2A5%29%2F%28sqrt%282%5E2%2B3%5E2%29%2Asqrt%284%5E2%2B5%5E2%29%29 = 23%2F%28sqrt%2813%29%2Asqrt%2841%29%29 =~ 0.996 (approx.)
Hence, alpha = arccos%280.996%29 = 0.0867 rad = 4.97°.

Answer.  The angle  alpha  between the vectors  u = (2,3)  and  v = (4,5)  is equal to  4.97°.



Example 2
Find the angle  alpha  between the vectors  u = (3,4)  and  v = (5,12)  in a coordinate plane.

Solution
The cosines of the angle  alpha  between the vectors  u = (3,4)  and  v = (5,12)  is equal to   cos%28alpha%29 = %283%2A5+%2B+4%2A12%29%2F%28sqrt%283%5E2%2B4%5E2%29%2Asqrt%285%5E2%2B12%5E2%29%29 = 63%2F%28sqrt%2825%29%2Asqrt%28169%29%29 = 63%2F%285%2A13%29 = 0.969 (approx.)
Hence, alpha = arccos%280.969%29 = 0.249 rad = 14.26°.

Answer.  The angle  alpha  between the vectors  u = (3,4)  and  v = (5,12)  is equal to  14.26°.


For the reference see the lesson  Dot-product of vectors in a coordinate plane and the angle between two vectors  under the topic  Introduction to vectors, addition and scaling  of the section  Algebra-II  in this site.

My lessons on Dot-product in this site are
    - Introduction to dot-product
    - Formula for Dot-product of vectors in a plane via the vectors components
    - Dot-product of vectors in a coordinate plane and the angle between two vectors
    - Perpendicular vectors in a coordinate plane
    - Solved problems on Dot-product of vectors and the angle between two vectors
    - Properties of Dot-product of vectors in a coordinate plane
    - The formula for the angle between two vectors and the formula for cosines of the difference of two angles

There are short lessons of the  "HOW TO . . . "  type on Dot-product:
    - HOW TO find dot-product of two vectors in a plane
    - HOW TO find scalar product of two vectors in a coordinate plane
    - HOW TO find the angle between two vectors in a coordinate plane                                            (this lesson)
    - HOW TO prove that two vectors in a coordinate plane are perpendicular
    - HOW TO prove that a triangle in a coordinate plane is a right triangle
    - HOW TO check if a quadrilateral in a coordinate plane is a parallelogram
    - HOW TO check if a quadrilateral in a coordinate plane is a rectangle
    - HOW TO check if a quadrilateral in a coordinate plane is a rhombus
    - HOW TO check if a quadrilateral in a coordinate plane is a square

For the full list of my lessons on dot-product with short annotations see the file  OVERVIEW of lessons on Dot-product.

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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