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Angle Between Vectors Calculator

Calculator for calculating the angle between vectors, calculates the cosine of the angle, and calculates the value of the angle in degrees and radians. This online calculator gives a detailed solution to all stages of the calculation.




Dimension of space
Specify the form of the first vector
Specify the form of the second vector

Enter the coordinates of the first vector
a̅ = { ; }


Enter coordinates of vector b
b̅ = { ; }



How to Calculate the Angle Between Vectors Located in 2d and 3d Space

Example #1
Let's calculate the angle between vectors in 3D space. The coordinates of both vectors are given by points.
Coordinates of point A of vector AB: (5 ; 9)
Coordinates of point B of vector AB: (-2 ; 11)
Coordinates of point C of vector CD: (0 ; 12)
Coordinates of point D of vector CD: (-3 ; 1)
cos α = AB ⋅ CD
|AB| ⋅ |CD|

Step by step solution:

1) Calculate the modulus (length) of the first and second vectors:

|AB| =
(xB - xA)2 + (yB - yA)2
=
(-2 - 5)2 + (11 - 9)2
=
(-7)2 + 22
=
49 + 4
=
53
= 7.28010988928052
|CD| =
(xD - xC)2 + (yD - yC)2
=
(-3 - 0)2 + (1 - 12)2
=
(-3)2 + (-11)2
=
9 + 121
=
130
= 11.4017542509914

2) Calculate the product of the modules of vectors:

|AB| ⋅ |CD| =
53
⋅
130
=
6890

3) Calculate the coordinates of the first vector from two points A and B:

AB = {xB - xA  ; yB - yA} = {-2 - 5 ; 11 - 9} = {-7 ; 2}

4) Calculate the coordinates of the second vector from two points C and D:

CD = {xD - xC  ; yD - yC} = {-3 - 0 ; 1 - 12} = {-3 ; -11}

5) Calculate the scalar product of vectors: AB and CD

AB ⋅ CD = ABxCDx + AByCDy = -7 ⋅ (-3) + 2 ⋅ (-11) = 21 + (-22) = -1

6) Calculate the cosine of the angle between the vectors:

cos α = AB ⋅ CD =
|AB| ⋅ |CD|
-1 /
6890
= -0.0120473184147734

7) Calculate the value of the angle ∠α between the vectors:

∠α = 1.58284393664908 Radians
∠α = 90.6902771978651° Degrees


Example #2
Let's calculate the angle between vectors in two-dimensional space.
Vector coordinates a: (5 ; 9)
Vector coordinates b: (-1 ; 7)
cos α = a ⋅ b
|a| ⋅ |b|

Step by step solution:

1) Calculate the modulus (length) of the first and second vectors:

|a| =
ax2 + ay2
=
52 + 92
=
25 + 81
=
106
= 10.295630140987
|b| =
bx2 + by2
=
(-1)2 + 72
=
1 + 49
=
50
= 5
2
= 7.07106781186548

2) Calculate the product of the modules of vectors:

|a| ⋅ |b| =
106
⋅
50
=
5300

3) Calculate the scalar product of vectors: a and b

a ⋅ b = axbx + ayby = 5 ⋅ (-1) + 9 ⋅ 7 = -5 + 63 = 58

4) Calculate the cosine of the angle between the vectors:

cos α = a ⋅ b =
|a| ⋅ |b|
58 /
5300
= 0.796691270902396

5) Calculate the value of the angle ∠α between the vectors:

∠α = 0.648995558996501 Radians
∠α = 37.1847064532332° Degrees


Example #3
Let's calculate the angle between vectors in 3D space. The coordinates of both vectors are given by points.
Coordinates of point A of vector AB: (7; 0.2 ; 69)
Coordinates of point B of vector AB: (-1 ; 0 ; 2/8)
Coordinates of point C of vector CD: (-4 ; -6 ; 2)
Coordinates of point D of vector CD: (3 ; 0 ; 9)
cos α = AB ⋅ CD
|AB| ⋅ |CD|

Step by step solution:

1) Calculate the modulus (length) of the first and second vectors:

|AB| =
(xB - xA)2 + (yB - yA)2 + (zB - zA)2
=
(-1 - 7)2 + (0 - 0.2)2 + (2/8 - 69)2
=
(-8)2 + (-0.2)2 + (-275/4)2
=
64 + 0.04 + (75625/16)
=
1916241
400
=
69.2141784607749

|CD| =
(xD - xC)2 + (yD - yC)2 + (zD - zC)2
=
(3 - (-4))2 + (0 - (-6))2 + (9 - 2)2
=
72 + 62 + 72
=
49 + 36 + 49
=
134
= 11.5758369027902

2) Calculate the product of the modules of vectors:

|AB| ⋅ |CD| =
1916241/400
⋅
134
=
641940.735

3) Calculate the coordinates of the first vector from two points A and B:

AB = {xB - xA  ; yB - yA; zB - zA} = {-1 - 7 ; 0 - 0.2 ; 2/8 - 69} = {-8 ; -1/5 ; -275/4}

4) Calculate the coordinates of the second vector from two points C and D:

CD = {xD - xC  ; yD - yC; zD - zC} = {3 - (-4) ; 0 - (-6) ; 9 - 2} = {7 ; 6 ; 7}

5) Calculate the scalar product of vectors: AB and CD

AB ⋅ CD = ABxCDx + AByCDy + ABzCDz = -8 ⋅ 7 + (-1/5) ⋅ 6 + (-275/4) ⋅ 7 = -56 + (-6/5) + (-1925/4) = -10769/20 = -538.45

6) Calculate the cosine of the angle between the vectors:

cos α = AB ⋅ CD =
|AB| ⋅ |CD|
-538.45 /
641940.735
= -0.672044318228661

7) Calculate the value of the angle ∠α between the vectors:

∠α = 2.30776235411475 Radians
∠α = 132.225043009951° Degrees


Example #4
Let's calculate the angle between vectors in 3D space.
Vector coordinates a: (5 ; 1 ; 7)
Vector coordinates b: (2 ; 4 ; 6)
cos α = a ⋅ b
|a| ⋅ |b|

Step by step solution:

1) Calculate the modulus (length) of the first and second vectors:

|a| =
ax2 + ay2 + az2
=
52 + 12 + 72
=
25 + 1 + 49
=
75
= 5
3
= 8.66025403784439
|b| =
bx2 + by2 + bz2
=
22 + 42 + 62
=
4 + 16 + 36
=
56
= 2
14
= 7.48331477354788

2) Calculate the product of the modules of vectors:

|a| ⋅ |b| =
75
⋅
56
=
4200

3) Calculate the scalar product of vectors: a and b

a ⋅ b = axbx + ayby + azbz = 5 ⋅ 2 + 1 ⋅ 4 + 7 ⋅ 6 = 10 + 4 + 42 = 56

4) Calculate the cosine of the angle between the vectors:

cos α = a ⋅ b =
|a| ⋅ |b|
56 /
4200
= 0.864098759787715

5) Calculate the value of the angle ∠α between the vectors:

∠α = 0.527439299499548 Radians
∠α = 30.2200458106607° Degrees
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