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प्रश्न
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then
पर्याय
\[x^2 + y^2 + z^2 = r^2\]
\[x^2 + y^2 - z^2 = r^2\]
\[x^2 - y^2 + z^2 = r^2\]
\[z^2 + y^2 - x^2 = r^2\]
उत्तर
Given:
`x= r sin θ cos Φ,`
`y=r sinθ sinΦ `
`z= r cos θ`
Squaring and adding these equations, we get
`x^2+y^2+z^2=(r sinθ cosΦ )^2+(r sin θ sinΦ )^2+(r cos θ)^2`
`= x^2+y^2+z^2=r^2 sin^2θ cos^2Φ+r^2 sin^2θsin^2Φ+r^2 cos^2θ `
`=x^2+y^2+z^2=(r^2 sin^2θ cos^2Φ+r^2 sin^2 sin^2Φ)+r^2 cos^2Φ`
`=x^2+y^2+z^2=r^2sin^2θ(cos^2Φ+sin^2Φ)+r^2 cos^2Φ`
`=x^2+y^2+z^2=r^2 sin^2θ(1)+r^2 cos^2θ`
`=x^2+y^2+z^2=r^2 sin^2θ+r^2 cos^2θ`
`=x^2+y^2+z^2=r^2(sin^2θ+cos^2θ)`
`=x^2+y^2+z^2=r^2(1)`
`=x^2+y^2+z^2=r^2`
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