Problem 31 - Algebra and Trigonometry

Question

A description of a plane is given. Find an equation for the plane. The plane that crosses the \(x\) -axis where \(x=1,\) the \(y\) -axis where \(y=3,\) and the \(z\) -axis where \(z=4\)

Step-by-Step Solution

Answer

The plane equation is \(12x + 4y + 3z = 12\).

Step 1: Identify the Intercepts

We are given a plane that intersects the axes at points (1,0,0), (0,3,0), and (0,0,4). These are the intercepts of the plane on the x-axis, y-axis, and z-axis, respectively.

Step 2: Use the Intercept Form of a Plane Equation

For a plane intersecting the x-axis, y-axis, and z-axis at points (a,0,0), (0,b,0), and (0,0,c), the equation of the plane can be written using the intercept form. This form is \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\).

Step 3: Substitute Intercept Values into the Equation

Substituting the given intercept values a=1, b=3, and c=4 into the intercept form gives us: \(\frac{x}{1} + \frac{y}{3} + \frac{z}{4} = 1\).

Step 4: Simplify the Plane Equation

The simplified form of the plane equation becomes \(x + \frac{y}{3} + \frac{z}{4} = 1\). To eliminate fractions, multiply through by 12 (the least common multiple of 1, 3, and 4) to yield: \(12x + 4y + 3z = 12\).

Key Concepts

Intercept Form

When dealing with planes in three dimensions, the intercept form is a helpful approach to find the equation of a plane. This method utilizes the points where the plane cuts through the x, y, and z axes. These points are known as intercepts. Imagine a plane that slices through these axes and touches them at certain intervals.
The intercept form of the plane equation is expressed as \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \). Here, \( a \), \( b \), and \( c \) represent the x, y, and z intercepts of the plane. This symbolic representation describes how the plane interacts with the three-dimensional space.

• Symmetry Across Axes: It assumes that the plane intercepts all three axes, which is a unique scenario that gives a clear geometric interpretation.
• Simple Substitutions: To craft the actual equation, you simply substitute the given intercept values into the formula.

This form is especially useful as it provides a direct way to derive the plane's equation when these intercepts are known.

Equation of a Plane

The equation of a plane in 3D space is essential for defining its position and orientation. Planes, being flat surfaces, require a distinct formula to describe them mathematically. Understanding this equation is crucial to exploring geometric relationships in three-dimensional spaces.
In general, a plane can be described by the equation \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are constants that give the plane its particular slope and tilt. These coefficients typically relate to the normal vector of the plane, which is perpendicular to the plane itself.

• Intercept Form: As seen in our original exercise, using specific intercepts allows for the introduction of the intercept form which we convert into a linear equation in standard form.
• Understanding Coefficients: The coefficients of \( x \), \( y \), and \( z \) serve to orient the plane. A change in these values results in a different plane orientation.

This equation represents an infinite set of points that form a flat surface in the three-dimensional coordinate grid, and it is central to various applications like computer graphics, architectural design, or physics.

3D Coordinate System

The 3D coordinate system is an extension of the 2D system, adding depth to the existing width and height dimensions. It is essential for describing spatial orientations, positions, and distances in most scientific and engineering contexts. This system uses three axes, known as the x, y, and z axes, which intersect at the origin.
Each axis in the 3D space is orthogonal (perpendicular) to the others, and points are usually represented as \((x, y, z)\), indicating the respective distance along each axis from the origin.

• Visualizing Planes: In a 3D coordinate system, planes can be visualized as infinite sheets that spread out through the space, slicing across the axes at different intercepts.
• Applications: This system is widely applied in fields that require spatial representation, including mathematics, physics, and video game design.

Understanding the 3D coordinate system allows for effective manipulation of geometric objects, calculation of volumes, and definition of shapes within varied parts of the space. It is foundational to working with three-dimensional vectors, surfaces, and space partitions.