If a vector makes angles `alpha,beta,gammaw i t hO X ,O Ya n dO Z` respectively, prove that `sin^2alpha+sin^2beta+sin^2gamma=2.`

Explanation

Suppose, a vector `vec{O P}` makes an angle alpha, beta, gamma with `O X`, `O Y`, `O Z` respectively.

Then, direction cosines of the vector are given by

`l=cos alpha, m=cos beta and n=cos gamma`

Consider,

`sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma=1-cos ^{2} alpha+1-cos ^{2} beta+1-cos ^{2} gamma `
`=3-(cos ^{2} alpha+cos ^{2} beta+cos ^{2} gammaright) `

`=3-(l^{2}+m^{2}+n^{2}right) quad[because l^{2}+m^{2}+n^{2}=1right] `

`=3-1 quad `

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