# 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.

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 #1Let'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 Î± = | AB â‹… CD | |

|AB| â‹… |CD| |

Step by step solution:

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

|AB| =_{B}- x

_{A})

^{2}+ (y

_{B}- y

_{A})

^{2}

^{2}+ (11 - 9)

^{2}

^{2}+ 2

^{2}

|CD| =

_{D}- x

_{C})

^{2}+ (y

_{D}- y

_{C})

^{2}

^{2}+ (1 - 12)

^{2}

^{2}+ (-11)

^{2}

2) Calculate the product of the modules of vectors:

|AB| â‹… |CD| =3) Calculate the coordinates of the first vector from two points A and B:

**AB = {x _{B} - x_{A }; y_{B} - y_{A}} = {-2 - 5 ; 11 - 9} = {-7 ; 2}**

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

**CD = {x _{D} - x_{C }; y_{D} - y_{C}} = {-3 - 0 ; 1 - 12} = {-3 ; -11}**

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

AB â‹… CD = AB_{x}CD

_{x}+ AB

_{y}CD

_{y}= -7 â‹… (-3) + 2 â‹… (-11) = 21 + (-22) = -1

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

cos Î± = | AB â‹… CD | = |

|AB| â‹… |CD| |

-1 / 6890 = -0.0120473184147734 | ||

7) Calculate the value of the angle âˆ Î± between the vectors:

**âˆ Î±**= 1.58284393664908 Radians

**âˆ Î±**= 90.6902771978651Â° 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 Î± = | a â‹… b | |

|a| â‹… |b| |

Step by step solution:

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

|a| =_{x}

^{2}+ a

_{y}

^{2}

^{2}+ 9

^{2}

|b| =

_{x}

^{2}+ b

_{y}

^{2}

^{2}+ 7

^{2}

2) Calculate the product of the modules of vectors:

|a| â‹… |b| =3) Calculate the scalar product of vectors: a and b

a â‹… b = a_{x}b

_{x}+ a

_{y}b

_{y}= 5 â‹… (-1) + 9 â‹… 7 = -5 + 63 = 58

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

cos Î± = | a â‹… b | = |

|a| â‹… |b| |

58 / 5300 = 0.796691270902396 | ||

5) Calculate the value of the angle âˆ Î± between the vectors:

**âˆ Î±**= 0.648995558996501 Radians

**âˆ Î±**= 37.1847064532332Â° 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 Î± = | AB â‹… CD | |

|AB| â‹… |CD| |

Step by step solution:

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

|AB| =_{B}- x

_{A})

^{2}+ (y

_{B}- y

_{A})

^{2}+ (z

_{B}- z

_{A})

^{2}

^{2}+ (0 - 0.2)

^{2}+ (2/8 - 69)

^{2}

^{2}+ (-0.2)

^{2}+ (-275/4)

^{2}

1916241 | ||

400 |

= | ||

69.2141784607749 | ||

|CD| =

_{D}- x

_{C})

^{2}+ (y

_{D}- y

_{C})

^{2}+ (z

_{D}- z

_{C})

^{2}

^{2}+ (0 - (-6))

^{2}+ (9 - 2)

^{2}

^{2}+ 6

^{2}+ 7

^{2}

2) Calculate the product of the modules of vectors:

|AB| â‹… |CD| =3) Calculate the coordinates of the first vector from two points A and B:

**AB = {x _{B} - x_{A }; y_{B} - y_{A}; z_{B} - z_{A}} = {-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 = {x _{D} - x_{C }; y_{D} - y_{C}; z_{D} - z_{C}} = {3 - (-4) ; 0 - (-6) ; 9 - 2} = {7 ; 6 ; 7}**

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

AB â‹… CD = AB_{x}CD

_{x}+ AB

_{y}CD

_{y}+ AB

_{z}CD

_{z}= -8 â‹… 7 + (-1/5) â‹… 6 + (-275/4) â‹… 7 = -56 + (-6/5) + (-1925/4) = -10769/20 = -538.45

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

cos Î± = | AB â‹… CD | = |

|AB| â‹… |CD| |

-538.45 / 641940.735 = -0.672044318228661 | ||

7) Calculate the value of the angle âˆ Î± between the vectors:

**âˆ Î±**= 2.30776235411475 Radians

**âˆ Î±**= 132.225043009951Â° 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 Î± = | a â‹… b | |

|a| â‹… |b| |

Step by step solution:

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

|a| =_{x}

^{2}+ a

_{y}

^{2}+ a

_{z}

^{2}

^{2}+ 1

^{2}+ 7

^{2}

|b| =

_{x}

^{2}+ b

_{y}

^{2}+ b

_{z}

^{2}

^{2}+ 4

^{2}+ 6

^{2}

2) Calculate the product of the modules of vectors:

|a| â‹… |b| =3) Calculate the scalar product of vectors: a and b

a â‹… b = a_{x}b

_{x}+ a

_{y}b

_{y}+ a

_{z}b

_{z}= 5 â‹… 2 + 1 â‹… 4 + 7 â‹… 6 = 10 + 4 + 42 = 56

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

cos Î± = | a â‹… b | = |

|a| â‹… |b| |

56 / 4200 = 0.864098759787715 | ||

5) Calculate the value of the angle âˆ Î± between the vectors:

**âˆ Î±**= 0.527439299499548 Radians

**âˆ Î±**= 30.2200458106607Â° Degrees