Question

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then 

Options

• \[x^2 + y^2 + z^2 = r^2\]

• \[x^2 + y^2 - z^2 = r^2\]

• \[x^2 - y^2 + z^2 = r^2\]

• \[z^2 + y^2 - x^2 = r^2\] 

Answer 1

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`