On 1st April 2024, person 'X' purchased a machinery costing ₹ 65000 and spent ₹ 10000 on its installation. The estimated effective life of the…

Direction cosines describe the direction a line points in 3D space. If a line makes angles α, β, and γ with the x-axis, y-axis, and z-axis respectively, then the direction cosines are $(cos α, cos β, cos γ)$(cosα,cosβ,cosγ), typically written as $(l, m, n)$(l,m,n).

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"Equally inclined with the coordinate axes" means the line makes the same angle with all three axes.

$$α = β = γ = θ$$α=β=γ=θ

The direction cosines become $(cos θ, cos θ, cos θ)$(cosθ,cosθ,cosθ).

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Direction cosines satisfy the fundamental property:

$$l² + m² + n² = 1$$l2+m2+n2=1

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Since $l = m = n = cos θ$l=m=n=cosθ:

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

$$3(cos θ)² = 1$$3(cosθ)2=1

$$cos²θ = \dfrac{1}{3}$$cos2θ=31​

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

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The $±$± sign appears because $cos θ$cosθ can be positive or negative depending on whether the angle is acute or obtuse. Each direction cosine can independently be $+\dfrac{1}{\sqrt{3}}$+3​1​ or $-\dfrac{1}{\sqrt{3}}$−3​1​.

Therefore, the direction cosines are $\left(±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}\right)$(±3​1​,±3​1​,±3​1​).