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# 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}.

Solution: calculate dot product of vectors:

a·b = 3 · 4 + 4 · 3 = 12 + 12 = 24.

Calculate vectors magnitude:

|a| = √32 + 42 = √9 + 16 = √25 = 5
|b| = √42 + 32 = √16 + 9 = √25 = 5

Calculate 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}.

Solution: calculate dot product of vectors:

a·b = 5 · 7 + 1 · 5 = 35 + 5 = 40.

Calculate vectors magnitude:

|a| = √72 + 12 = √49 + 1 = √50 = 5√2
|b| = √52 + 52 = √25 + 25 = √50 = 5√2

Calculate 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}.

Solution: calculate dot product of vectors:

a·b = 3 · 4 + 4 · 4 + 0 · 2 = 12 + 16 + 0 = 28.

Calculate vectors magnitude:

|a| = √32 + 42 + 02 = √9 + 16 = √25 = 5
|b| = √42 + 42 + 22 = √16 + 16 + 4 = √36 = 6

Calculate 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}.

Solution: calculate dot product of vectors:

a·b = 1 · 5 + 0 · 5 + 3 · 0 = 5.

Calculate vectors magnitude:

|a| = √12 + 02 + 32 = √1 + 9 = √10
|b| = √52 + 52 + 02 = √25 + 25 = √50 = 5√2

Calculate the angle between vectors:

 cos α = a · b = 5 = 1 = √5 = 0.1√5 |a| · |b| √10 · 5√2 2√5 10

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