[x^2 + (y - 1)^2 + z^2] - [(x - 1)^2 + y^2 + z^2] = 0
![[x^2 + (y - 1)^2 + z^2] - [(x - 1)^2 + y^2 + z^2] = 0](https://soloferat.biz.id/images/x2--y---12--z2---x---12--y2--z2--0.jpg)
Understanding the Equation: x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
The equationx² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0might look complex at first glance, but it represents a meaningful geometric relationship in three-dimensional space. This article breaks down the equation, simplifies it, explores its geometric interpretation, and explains its relevance in applied mathematics and problem-solving.
Simplifying the Equation
Start with:x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
First, expand each term carefully:
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Expand (y - 1)²:(y - 1)² = y² - 2y + 1 So, x² + y² - 2y + 1 + z²
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Expand (x - 1)²:(x - 1)² = x² - 2x + 1 So, (x² - 2x + 1) + y² + z²
Now substitute both into the original expression:
\[[x² + y² - 2y + 1 + z²] - [x² - 2x + 1 + y² + z²] = 0\]
Distribute the minus sign:
\[x² + y² - 2y + 1 + z² - x² + 2x - 1 - y² - z² = 0\]
Now combine like terms:
- \(x² - x² = 0\)- \(y² - y² = 0\)- \(z² - z² = 0\)- \(-2y + 2x + 1 - 1 = 2x - 2y\)
So the simplified equation is:2x - 2y = 0which further reduces to:x – y = 0or equivalently:x = y
Geometric Interpretation
The simplified equation x = y describes a plane in three-dimensional space where the x-coordinate equals the y-coordinate. This is a vertical plane that slices through all values of z, passing diagonally across the xy-plane along the line where x = y.
Visualize shifting the classic planes. Instead of aligning with coordinate axes, this plane cuts diagonally from the origin along where x equals y, forming a “square-like” diagonal across quadrants where x and y are equal in magnitude and sign.
This plane is fundamental in symmetry considerations and acts as a decision boundary in optimization, machine learning, and physics problems involving diagonal variation or equilibrium.
Applications and Relevance
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Geometry & Graphing: This equation helps students and graphics programmers define lines or planes where x and y share proportional relationships — useful in 3D modeling and geometric constraints.
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Physics & Symmetry: In coordinate systems aligned with physical symmetry (especially when x and y vary equally), such equations simplify problem setups in statics, fields, or wave propagation.
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Machine Learning & Optimization: The simplified form x = y resembles binary classification or decision boundaries where two features are balanced — valuable in training models or formulating constraints in convex optimization.
Summary
- The equation x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0 simplifies to the plane x = y.- It represents a diagonal slicing plane in 3D space where x and y coordinates are equal.- Useful in geometry, optimization, physics, and machine learning due to its symmetry and constraint-forming nature.
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Understanding this equation unlocks deeper insight into spatial relationships defined by differences in coordinates — a building block for advanced mathematical modeling and algorithmic logic.
