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Question
Evaluate `|(cos alpha cos beta, cos alpha sin beta, -sin alpha),(-sin beta, cos beta, 0),(sin alpha cos beta, sin alpha sin beta,cos alpha )|`
Solution
Δ = `[(cosalphacosbeta, cosalphasinbeta,-sinalpha),(-sinbeta,cosbeta,0),(sinalpha cosbeta,sinalphasinbeta,cosalpha)]`
Expanding along C3,
`= cos alphacosbeta |(cosbeta, 0), (sinalphasinbeta,cosalpha)| - cosalpha sinbeta |(-sinbeta, 0), (sinalphacosbeta, cosalpha)| - sinalpha|(-sinbeta, cosbeta), (sinalpha cosbeta, sin alpha sin beta)|`
we have:
Δ = `-sinalpha(-sinalphasin^2beta - cos^2betasinalpha) + cosalpha(cosalphacos^2beta + cosalphasin^2beta)`
= `sin^2alpha(sin^2beta + cos^2beta) + cos^2alpha(cos^2beta + sin^2beta)`
= `sin^2alpha(1) + cos^2alpha(1)`
= 1
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