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Question

If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin2 α + sin2 β + sin2 γ.

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Solution

Suppose, a vector OP makes an angle α, β, γ with OX, OY, OZ respectively. Then, direction cosines of the vector are given by
l = cos α , m = cos β and n = cos γ
Consider,
sin2α + sin2 β + sin2 γ = 1 - cos2α + 1 - cos2β + 1- cos2γ = 3 - cos2α + cos2β + cos2γ = 3 - l2+m2+n2
= 3 - 1 [∵ l2 + m2 + n2 = 1]
= 2

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