A trust invites a deposit of lumpsum amount from individual so that annual scholarship of Rs.5000 is paid. Rate of interest is 5% per annum. If the…

Direction cosines are the cosines of the angles that a line makes with the x, y, and z axes. If a line makes angles α, β, and γ with the three axes, the direction cosines are:

$l = \cos α$l=cosα (angle with x-axis)

$m = \cos β$m=cosβ (angle with y-axis)

$n = \cos γ$n=cosγ (angle with z-axis)

These satisfy the relation: $l^2 + m^2 + n^2 = 1$l2+m2+n2=1

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The line makes equal angles with all three coordinate axes, so:

$$α = β = γ = θ$$α=β=γ=θ

The direction cosines become:

$$l = \cos θ$$l=cosθ

$$m = \cos θ$$m=cosθ

$$n = \cos θ$$n=cosθ

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Using $l^2 + m^2 + n^2 = 1$l2+m2+n2=1:

$$(\cos θ)^2 + (\cos θ)^2 + (\cos θ)^2 = 1$$(cosθ)2+(cosθ)2+(cosθ)2=1

$$3 \cos^2 θ = 1$$3cos2θ=1

$$\cos^2 θ = \dfrac{1}{3}$$cos2θ=31​

$$\cos θ = ±\dfrac{1}{\sqrt{3}}$$cosθ=±3​1​

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Since $l = m = n = \cos θ$l=m=n=cosθ:

Direction cosines $= \left(±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}\right)$=(±3​1​,±3​1​,±3​1​)

The $±$± sign accounts for the line pointing in two opposite directions.