Answer:
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.
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.