Problem 75 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The planes are those in 3D space represented by equations of the form \( Ax + By + Cz = D \) where each term \( A, B, C, D \) relates to \( a, b, c \).

Step 1: Understanding the Equation

The equation \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \) represents a plane in a three-dimensional space. The plane is defined for any non-zero constants \( a, b, \) and \( c \). Here, \( x, y, \) and \( z \) are the variables representing the coordinates in this 3D space.

Step 2: Rewriting the Equation

The given equation can be rewritten as \( x/a + y/b + z/c = 1 \). Multiplying through by the common denominator \( abc \) results in the equation \( bcx + acy + abz = abc \), which is a standard form of the equation of a plane: \( Ax + By + Cz = D \).

Step 3: Comparing with Standard Plane Equation

In the rewritten equation \( bcx + acy + abz = abc \), the coefficients \( A, B, \) and \( C \) correspond to \( bc, ac, \) and \( ab \), respectively, and \( D \) is equal to \( abc \). This confirms that the given equation indeed describes a plane for any non-zero \( a, b, \) and \( c \).

Step 4: Identifying Specific Planes

The types of planes described by the equation \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \) include any planes where the sum of the fractions of the coordinates divided by their respective constants equals 1. Each specific choice of \( a, b, \) and \( c \) corresponds to a unique plane.

Key Concepts

Three-Dimensional Space

In mathematics, three-dimensional space refers to a geometric setting in which three values are needed to determine the position of an element. This space is commonly represented with the coordinates

• x: Horizontal axis
• y: Vertical axis
• z: Depth axis

Imagine a box, where:

• x is the left-to-right direction
• y is the up-and-down direction
• z is the front-to-back direction

This concept is crucial in geometry and physics, as it provides a framework for modeling objects and phenomena in the real world. Each point in the three-dimensional space can be described by a set of three coordinates: \( (x, y, z) \). Hence, a plane in this space is defined by an equation involving these three variables.

Standard Form of a Plane

The standard form of a plane equation is expressed as \( Ax + By + Cz = D \), where \( A, B, \) and \( C \) are coefficients and \( D \) is the constant term. This form allows mathematicians to describe any plane. Furthermore, any point \( (x, y, z) \) that satisfies this equation lies on the plane.
Given a specific plane equation such as \[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \], we can manipulate it into the standard form by eliminating the fractions. We achieve this by multiplying each term by the common denominator, resulting in \[ bcx + acy + abz = abc \]. This is in the standard form, making it clear how the plane relates to the x, y, and z coordinates.

Coefficients in a Plane Equation

The coefficients \( A, B, \) and \( C \) in a plane equation \( Ax + By + Cz = D \) have significant roles. They determine the orientation and tilt of the plane within the three-dimensional space.
In the rewritten plane equation \[ bcx + acy + abz = abc \], the coefficients are defined as:

• \( A = bc \)
• \( B = ac \)
• \( C = ab \)

These values link directly back to the constants \( a, b, \) and \( c \) in the original equation \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \). By selecting different non-zero values for \( a, b, \) and \( c \), we can define different planes, each uniquely oriented and positioned in the three-dimensional space.