Answer:
The direction cosines are:
 ,
,  and
  and   with respect to the x, y and z axes respectively.
  with respect to the x, y and z axes respectively.
The direction angles are:
 40°,  81° and  52° with respect to the x, y and z axes respectively.
Step-by-step explanation:
For a given vector a = ai + aj + ak, its direction cosines are the cosines of the angles which it makes with the x, y and z axes.
If a makes angles α, β, and γ (which are the direction angles) with the x, y and z axes respectively, then its direction cosines are: cos α, cos β and cos γ in the x, y and z axes respectively.
Where;
cos α =  ---------------------(i)
               ---------------------(i)
cos β =  ---------------------(ii)
               ---------------------(ii)
cos γ =  ----------------------(iii)
             ----------------------(iii)
<em>And from these we can get the direction angles as follows;</em>
α =  cos⁻¹ (  )
 )
β = cos⁻¹ (  )
 )
γ = cos⁻¹ (  )
 )
Now to the question:
Let the given vector be
a = 5i + j + 4k
a . i =  (5i + j + 4k) . (i)
a . i = 5         [a.i <em>is just the x component of the vector</em>]
a . j = 1            [<em>the y component of the vector</em>]
a . k = 4          [<em>the z component of the vector</em>]
<em>Also</em>
|a|. |i| = |a|. |j| = |a|. |k| = |a|           [since |i| = |j| = |k| = 1]
|a| = 
|a| = 
|a| = 
Now substitute these values into equations (i) - (iii) to get the direction cosines. i.e
cos α = 
cos β =   
              
cos γ =   
 
From the value, now find the direction angles as follows;
α =  cos⁻¹ (  )
 )
α =  cos⁻¹ (  )
 )
α =  cos⁻¹ ( )
 )
α =  cos⁻¹ (0.7715)
α = 39.51
α = 40°
β = cos⁻¹ (  )
 )
β = cos⁻¹ (  )
 )
β = cos⁻¹ (  )
 )
β = cos⁻¹ ( 0.1543 )
β = 81.12
β = 81°
γ = cos⁻¹ (  )
 )
γ = cos⁻¹ ( )
)
γ = cos⁻¹ ( )
)
γ = cos⁻¹ (0.6172)
γ = 51.89
γ = 52°
<u>Conclusion:</u>
The direction cosines are:
 ,
,  and
  and   with respect to the x, y and z axes respectively.
  with respect to the x, y and z axes respectively.
The direction angles are:
 40°,  81° and  52° with respect to the x, y and z axes respectively.