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If x = r sinθ cos φ, y = r sinθ sinφ and z = r cosθ, then _________.
- a)x2 + y2 + z2 = r2
- b)x2 + y2 – z2 = r2
- c)x2 – y2 + z2 = r2
- d)z2 + y2 – x2 = r2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If x = r sinθ cos φ, y = r sinθ sinφ and z = r cos...
x = r sinθcosφ ... (i)
y = r sinθsinφ ... (ii)
z = r cosθ ... (iii)
Squaring and adding (i) and (ii), we get
x2 + y2 = r2sin2q ... (iv)
Squaring (iii) and adding it with (iv), we get
x2 + z2 + y2 = r2
y = r sinθsinφ ... (ii)
z = r cosθ ... (iii)
Squaring and adding (i) and (ii), we get
x2 + y2 = r2sin2q ... (iv)
Squaring (iii) and adding it with (iv), we get
x2 + z2 + y2 = r2
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Community Answer
If x = r sinθ cos φ, y = r sinθ sinφ and z = r cos...
Understanding the Given Equations
The equations provided represent the transformation from spherical coordinates to Cartesian coordinates. Here, r is the radius, θ (theta) is the polar angle, and φ (phi) is the azimuthal angle.
Equations Explained
- x = r sin θ cos φ
- y = r sin θ sin φ
- z = r cos θ
These equations describe how to express a point in three-dimensional space using spherical coordinates.
Deriving x² + y² + z²
To verify the correct answer, we will calculate x² + y² + z²:
1. Calculate x²:
- x² = (r sin θ cos φ)² = r² sin² θ cos² φ
2. Calculate y²:
- y² = (r sin θ sin φ)² = r² sin² θ sin² φ
3. Calculate z²:
- z² = (r cos θ)² = r² cos² θ
4. Combine the Results:
- x² + y² + z² = r² sin² θ cos² φ + r² sin² θ sin² φ + r² cos² θ
- Factoring out r²:
- = r² [sin² θ (cos² φ + sin² φ) + cos² θ]
- Since (cos² φ + sin² φ) = 1, we get:
- = r² [sin² θ + cos² θ]
- Again, since (sin² θ + cos² θ) = 1, this simplifies to:
- = r²
Conclusion
Thus, the expression x² + y² + z² equals r², confirming that option 'A' is indeed the correct answer. This relationship is fundamental in connecting spherical coordinates with Cartesian coordinates.
The equations provided represent the transformation from spherical coordinates to Cartesian coordinates. Here, r is the radius, θ (theta) is the polar angle, and φ (phi) is the azimuthal angle.
Equations Explained
- x = r sin θ cos φ
- y = r sin θ sin φ
- z = r cos θ
These equations describe how to express a point in three-dimensional space using spherical coordinates.
Deriving x² + y² + z²
To verify the correct answer, we will calculate x² + y² + z²:
1. Calculate x²:
- x² = (r sin θ cos φ)² = r² sin² θ cos² φ
2. Calculate y²:
- y² = (r sin θ sin φ)² = r² sin² θ sin² φ
3. Calculate z²:
- z² = (r cos θ)² = r² cos² θ
4. Combine the Results:
- x² + y² + z² = r² sin² θ cos² φ + r² sin² θ sin² φ + r² cos² θ
- Factoring out r²:
- = r² [sin² θ (cos² φ + sin² φ) + cos² θ]
- Since (cos² φ + sin² φ) = 1, we get:
- = r² [sin² θ + cos² θ]
- Again, since (sin² θ + cos² θ) = 1, this simplifies to:
- = r²
Conclusion
Thus, the expression x² + y² + z² equals r², confirming that option 'A' is indeed the correct answer. This relationship is fundamental in connecting spherical coordinates with Cartesian coordinates.
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