Problem 71 - Thomas Calculus

Question

For any nonzero numbers \(a, b,\) and \(c,\) the graph of \((x / a)+\) \((y / b)+(z / c)=1\) is a plane. Which planes have an equation of this form?

Step-by-Step Solution

Answer

Planes with equation \( Ax + By + Cz = D \), where \( A, B, C \) are nonzero.

Step 1: Identify the General Form of the Equation

The given equation \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \) is the equation of a plane in three-dimensional space. In general, any equation of the form \( Ax + By + Cz = D \) represents a plane where \( A, B, C, \) and \( D \) are constants.

Step 2: Rewrite the Equation in Standard Form

To match the form \( Ax + By + Cz = D \), we first find a common denominator for the left hand side of the equation: \( \frac{bxcz}{abc} + \frac{ayc}{abc} + \frac{xab}{abc} = 1 \) becomes \( bcx + acy + abz = abc \). This matches the standard plane equation \( Ax + By + Cz = D \), where \( A = bc \), \( B = ac \), \( C = ab \), and \( D = abc \).

Step 3: Generalize the Plane Equation in Context

The equation \( bcx + acy + abz = abc \) shows that any plane in three-dimensional space can be expressed as a combination of variables with constants following the adjusted form \( Ax + By + Cz = D \) where \( A, B, \) and \( C \) are nonzero constants and \( D \) is a constant representing the particular position of the plane.

Key Concepts

Understanding Three-Dimensional Space

Three-dimensional space is where our world exists, and it's defined by three axes: the x-axis, y-axis, and z-axis. Together, these axes allow us to pinpoint a location with coordinates in the form of \(x, y, z\). In such a space, every point, object, or plane has a specific position derived from these coordinates. This can be likened to finding your exact spot in a room by knowing your distance from three walls.
In mathematical terms, this means locating a point using three numbers instead of two, as you would in two-dimensional or flat space. These three coordinates give every point a unique position in the three-dimensional universe.
By understanding this, we can better appreciate how planes and other shapes are described in three-dimensional spaces. In essence, planes are flat, two-dimensional surfaces extended infinitely in the space defined by these three axes.

Connecting with Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors and linear equations. It is the foundation for understanding three-dimensional spaces through concepts like vector addition, multiplication, and transformation. When it comes to planes, linear algebra provides the tools to describe, analyze, and manipulate them using equations involving variables and constants.
One of the central ideas in linear algebra is the plane equation, which is used to describe the position and orientation of a plane in space. By understanding how linear equations like \(Ax + By + Cz = D\) work, we can better handle problems involving planes.
Here are some of the key concepts linear algebra brings to planes:

• **Vectors:** Represent directions and magnitudes in space.
• **Matrices:** Act as transformations for rotating or moving planes.
• **Scalar Multiplication:** Extends vectors in planes to create scalable equations.

With these concepts, you can approach vectors and planes analytically and gain deeper insights into their properties.

Decoding the Plane Equation

A plane equation in three-dimensional space helps us define any flat surface. For example, the formula \(Ax + By + Cz = D\) clearly indicates a plane by assigning specific roles to the constants and variables. Here’s how it breaks down:
- **Variables \(x, y, z\):** Represent the dimensions set by the spatial axes.
- **Constants \(A, B, C\):** Define the plane’s orientation in space by acting as coefficients for each axis.
- **Constant \(D\):** Establishes the plane's unique position by setting a specific value for the equation.
By utilizing these elements, a plane can essentially slice through three-dimensional space at different angles and positions.
Moreover, the constants \(A, B, C\) must not all be zero; otherwise, the equation would no longer represent a plane. Remember, understanding these elements provides a solid foundation for exploring more complex geometrical shapes and surfaces.