# Direction cosines of a vector

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## Direction cosines of a vector - definition

Definition. The

**direction cosines of the vector**a are the cosines of angles that the vector forms with the coordinate axes.The direction cosines uniquely set the direction of vector.

Basic relation. To find the

**direction cosines of the vector**a is need to divided the corresponding coordinate of vector by the length of the vector.The coordinates of the unit vector is equal to its direction cosines.

Property of direction cosines. The sum of the squares of the direction cosines is equal to one.

## Direction cosines of a vector formulas

### Direction cosines of a vector formula for two-dimensional vector

In the case of the plane problem (Fig. 1) the direction cosines of a vector a = {a_{x} ; a_{y}} can be found using the following formula

cos α = | a_{x} | ; | cos β = | a_{y} |

|a| | |a| |

**Property:**

cos

^{2}α + cos

^{2}β = 1

Fig. 1 |

### Direction cosines of a vector formula for three-dimensional vector

In the case of the spatial problem (Fig. 2) the direction cosines of a vector a = {a_{x} ; a_{y} ; a_{z}} can be found using the following formula

cos α = | a_{x} | ; | cos β = | a_{y} | ; | cos γ = | a_{z} |

|a| | |a| | |a| |

**Property:**

cos

^{2}α + cos^{2}β + cos^{2}γ = 1Fig. 2 |

## Examples of tasks

### Examples of plane tasks

Example 1. Find the direction cosines of the vector a = {3; 4}.

|a| = √3

Calculate the direction cosines of the vector a:

**Solution:**

|a| = √3

^{2}+ 4^{2}= √9 + 16 = √25 = 5.Calculate the direction cosines of the vector a:

cos α = | a_{x} | = | 3 | = 0.6 |

|a| | 5 |

cos β = | a_{y} | = | 4 | = 0.8 |

|a| | 5 |

**Answer:** direction cosines of the vector a is cos α = 0.6, cos β = 0.8.

Example 2. Find the vector a if it length equal to 26, and direction cosines is cos α = 5/13, cos β = -12/13.

a

**Solution:**

_{x}= |a| · cos α = 26 · 5/13 = 10a

_{y}= |a| · cos β = 26 · (-12/13) = -24**Answer:** a = {10; -24}.

### Examples of spatial tasks

Example 3. Find the direction cosines of the vector a = {2; 4; 4}.

|a| = √2

Calculate the direction cosines of the vector a:

1 3 , cos β = 2 3 , cos γ = 2 3 .

**Solution:**

|a| = √2

^{2}+ 4^{2}+ 4^{2}= √4 + 16 + 16 = √36 = 6.Calculate the direction cosines of the vector a:

cos α = | a_{x} | = | 2 | = | 1 |

|a| | 6 | 3 |

cos β = | a_{y} | = | 4 | = | 2 |

|a| | 6 | 3 |

cos γ = | a_{z} | = | 4 | = | 2 |

|a| | 6 | 3 |

**Answer:**direction cosines of the vector a is cos α =
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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