If l , m , n are the direction cosines of a line which makes angles α , β and γ with the coordinate axes X , Y , Z , respectively, then l m + m n + n l takes the maximum value when

Question

If l , m , n are the direction cosines of a line which makes angles α , β and γ with the coordinate axes X , Y , Z , respectively, then l m + m n + n l takes the maximum value when, α , β , γ are in arithmetic progression, α = β = γ, any two of α , β , γ are the same, one of α , β , γ is zero and the remaining two are non-zero and unequal.

MARKS App · Accepted Answer

As we know, l = cos α m = cos β n = cos γ Also, cos 2 α + cos 2 β + cos 2 γ = 1 Now l m + m n + n l = cos α cos β + cos β cos γ + cos γ cos α l m + m n + n l will take the maximum value only when α = β = γ , that will be l m + m n + n l = 3 cos 2 α = 1

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