If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin²α + sin²β + sin²γ.

Question

If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin 2 α + sin 2 β + sin 2 γ.

Philoid · Accepted Answer

The sum of squares of direction cosines of a vector is 1.

Let the angles made by vector be α, β, γ. Then, we get

cos²α+cos²β+cos²γ=1

using cos²θ=1-sin²θ, we get

(1-sin²α) +(1-sin²β) +(1-sin²γ) =1

Or, sin²α+sin²β+sin²γ=2

If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin2α + sin2β + sin2γ.