o paint a house, two painters are hired. The first painter can paint the entire house in ten hours. The second painter will need
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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.
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)
<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
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.