Angle Between Two Vectors - OnlineMSchool

Angle between two vectorsPage Navigation:

• Angle between two vectors - definition
• Angle between two vectors - formula
• Examples of tasks
  • plane tasks
  • spatial tasks

Online calculator. 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. 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

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 Online calculators with vectors Tasks and exercises with vector 2D Tasks and exercises with vector 3D

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