[x^2 + (y - 1)^2 + z^2] - [(x - 1)^2 + y^2 + z^2] = 0 - Groen Casting
Understanding the Equation: x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
Understanding the Equation: x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
The equation
x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
might 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.
Understanding the Context
Simplifying the Equation
Start with:
x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
First, expand each term carefully:
- Expand (y - 1)²:
(y - 1)² = y² - 2y + 1
So, x² + y² - 2y + 1 + z²
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Key Insights
- 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:
\[
x² + y² - 2y + 1 + z² - x² + 2x - 1 - y² - z² = 0
\]
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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 = 0
which further reduces to:
x – y = 0
or 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.
