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 (cosα,cosβ,cosγ), typically written as (l,m,n).
"Equally inclined with the coordinate axes" means the line makes the same angle with all three axes.
α=β=γ=θ
The direction cosines become (cosθ,cosθ,cosθ).
Direction cosines satisfy the fundamental property:
l2+m2+n2=1
Since l=m=n=cosθ:
(cosθ)2+(cosθ)2+(cosθ)2=1
3(cosθ)2=1
cos2θ=31
cosθ=±31
The ± sign appears because cosθ can be positive or negative depending on whether the angle is acute or obtuse. Each direction cosine can independently be +31 or −31.
Therefore, the direction cosines are (±31,±31,±31).