%5E3%2F27%3D(y-2)%5E3%2F8%3D(z-3)%5E3%2F64.jpeg "(x-1)^3/27=(y-2)^3/8=(z-3)^3/64")
Exploring the Equation (x-1)³/27 = (y-2)³/8 = (z-3)³/64
This equation represents a fascinating geometric concept, specifically a parametric equation of a line. Let's delve into understanding its properties and significance.
Understanding the Structure
The equation is formed by equating three ratios:
- (x-1)³/27: This represents a cube of the form (x-1), scaled down by a factor of 3³.
- (y-2)³/8: This represents a cube of the form (y-2), scaled down by a factor of 2³.
- (z-3)³/64: This represents a cube of the form (z-3), scaled down by a factor of 4³.
The key observation is that each of these cubes is scaled differently, hinting at a proportional relationship between the changes in x, y, and z.
Geometric Interpretation
The equation describes a straight line in three-dimensional space. Here's why:
- Direction: The scaling factors (3, 2, 4) in the denominators dictate the direction of the line. These factors represent the slope of the line with respect to the x, y, and z axes.
- Point: The constants (1, 2, 3) in the numerators indicate a specific point on the line. This point, (1, 2, 3), serves as a starting point for the line's trajectory.
Parameterization
We can rewrite the equation in a more explicit form using a parameter 't':
(x - 1)/3 = (y - 2)/2 = (z - 3)/4 = t
This parameterized form allows us to express the coordinates of any point on the line in terms of the parameter 't':
- x = 3t + 1
- y = 2t + 2
- z = 4t + 3
As 't' varies, the equations generate different points along the straight line.
Applications
The equation, and its parameterized form, find applications in various fields:
- Geometry: Understanding lines in 3D space is fundamental to various geometric calculations and constructions.
- Vector Analysis: The direction ratios (3, 2, 4) can be interpreted as a direction vector representing the line's orientation in space.
- Physics: Parametric equations of lines are used to model the trajectory of objects moving in a straight path.
Summary
The equation (x-1)³/27 = (y-2)³/8 = (z-3)³/64 represents a straight line in 3D space. It provides a concise description of the line's direction and a point it passes through. Its parameterized form allows for the efficient generation of points along the line, making it a valuable tool in various mathematical and scientific applications.