# Angle between two vectors

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Definition. The

**angle between two vectors**, deferred by a single point, called the shortest angle at which you have to turn around one of the vectors to the position of co-directional with another vector.Basic relation. The

**cosine of the angle between two vectors**is equal to the dot product of this vectors divided by the product of vector magnitude.## Angle between two vectors - formula

cos α = | a·b |

|a|·|b| |

## Examples of tasks

### Examples of plane tasks

Example 1. Find the angle between two vectors a = {3; 4} and b = {4; 3}.

|b| = √4

**Solution:** calculate dot product of vectors:

Calculate vectors magnitude:

|a| = √3^{2}+ 4^{2}= √9 + 16 = √25 = 5|b| = √4

^{2}+ 3^{2}= √16 + 9 = √25 = 5Calculate the angle between vectors:

cos α = | a · b | = | 24 | = | 24 | = 0.96 |

|a| · |b| | 5 · 5 | 25 |

Example 2. Find the angle between two vectors a = {7; 1} and b = {5; 5}.

|b| = √5

**Solution:** calculate dot product of vectors:

Calculate vectors magnitude:

|a| = √7^{2}+ 1^{2}= √49 + 1 = √50 = 5√2|b| = √5

^{2}+ 5^{2}= √25 + 25 = √50 = 5√2Calculate the angle between vectors:

cos α = | a · b | = | 40 | = | 40 | = | 4 | = 0.8 |

|a| · |b| | 5√2 · 5√2 | 50 | 5 |

### Examples of spatial tasks

Example 3. Find the angle between two vectors a = {3; 4; 0} and b = {4; 4; 2}.

|b| = √4

**Solution:** calculate dot product of vectors:

Calculate vectors magnitude:

|a| = √3^{2}+ 4^{2}+ 0^{2}= √9 + 16 = √25 = 5|b| = √4

^{2}+ 4^{2}+ 2^{2}= √16 + 16 + 4 = √36 = 6Calculate the angle between vectors:

cos α = | a · b | = | 28 | = | 14 |

|a| · |b| | 5 · 6 | 15 |

Example 4. Find the angle between two vectors a = {1; 0; 3} and b = {5; 5; 0}.

|b| = √5

**Solution:** calculate dot product of vectors:

Calculate vectors magnitude:

|a| = √1^{2}+ 0^{2}+ 3^{2}= √1 + 9 = √10|b| = √5

^{2}+ 5^{2}+ 0^{2}= √25 + 25 = √50 = 5√2Calculate the angle between vectors:

cos α = | a · b | = | 5 | = | 1 | = | √5 | = 0.1√5 |

|a| · |b| | √10 · 5√2 | 2√5 | 10 |

Vectors
Vectors Definition. Main information
Component form of a vector with initial point and terminal point
Length of a vector
Direction cosines of a vector
Equal vectors
Orthogonal vectors
Collinear vectors
Coplanar vectors
Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalar-vector multiplication
Dot product of two vectors
Cross product of two vectors (vector product)
Scalar triple product (mixed product)
Linearly dependent and linearly independent vectors
Decomposition of the vector in the basis

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