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Angle between two vectors

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.
Angle between two vectors
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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