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# 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 = {ax ; ay} can be found using the following formula

 cos α = ax ; cos β = ay |a| |a|

Property:

cos
2 α + cos2 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 = {ax ; ay ; az} can be found using the following formula

 cos α = ax ; cos β = ay ; cos γ = az |a| |a| |a|

Property:

cos2 α + cos2 β + cos2 γ = 1 Fig. 2

## Examples of tasks

### Examples of plane tasks

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

Solution:

Calculate the length of vector a:
|a| = √32 + 42 = √9 + 16 = √25 = 5.

Calculate the direction cosines of the vector a:
 cos α = ax = 3 = 0.6 |a| 5
 cos β = ay = 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.

Solution:

ax = |a| · cos α = 26 · 5/13 = 10
ay = |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}.

Solution:

Calculate the length of vector a:
|a| = √22 + 42 + 42 = √4 + 16 + 16 = √36 = 6.

Calculate the direction cosines of the vector a:
 cos α = ax = 2 = 1 |a| 6 3
 cos β = ay = 4 = 2 |a| 6 3
 cos γ = az = 4 = 2 |a| 6 3

Answer: direction cosines of the vector a is cos α = 13, cos β = 23, cos γ = 23.

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