Direction cosines of a vector

Definition. The direction cosines of the vector a are the cosines of angles that the vector forms with the coordinate axes.

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.

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

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

Property:

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

Property:

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:

a_x = |a| · cos α = 26 · 5/13 = 10
a_y = |a| · cos β = 26 · (-12/13) = -24

Answer: a = {10; -24}.

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.