Question

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

A. x² + y² + z² = r² 

B. x² + y²− z² = r² 

C. x²− y² + z² = r² 

D. z² + y²− x² = r²

Answer 1

Given: x = r sin θ cos ϕ, y = r sin θ sin φ and z = r cos θ, 

Solution: x = r sin θ cos ϕ 

Squaring both sides, we get 

x² = r² sin² θ cos²ϕ …….(i) 

and y = r sin θ sin ϕ

Squaring both sides, we get 

⇒ y² = r² sin² θ sin²ϕ …….(ii) 

z = r cos θSquaring both sides, we get

⇒ z² = r² cos²θ ….(iii) 

Adding (i), (ii) and (iii), we get 

x² + y² + z² = r² sin² θ cos²ϕ + r² sin²θ sin²ϕ + r² cos²θ 

= r² (sin²θ cos²ϕ + sin²θ sin²ϕ + cos²θ) 

= r² [sin²θ (cos²ϕ + sin²ϕ) + cos²θ] 

∵ sin²θ + cos²θ = 1 

= r² [sin²θ + cos²θ] 

Again apply the identity sin²θ + cos²θ = 1

= r² 

Hence x² + y² + z² = r²