Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 5, 1, 4

The direction cosines are:

,  and    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)

cos β =               ---------------------(ii)

cos γ =             ----------------------(iii)

And from these we can get the direction angles as follows;

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 is just the x component of the vector]

a . j = 1            [the y component of the vector]

a . k = 4          [the z component of the vector]

Also

|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°

Conclusion:

The direction cosines are:

,  and    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.