Problem 66 - Advanced Engineering Mathematics

Question

In Problems, find parametric equations for the line of intersection of the given planes. $$ \begin{array}{r} x+2 y-z=2 \\ 3 x-y+2 z=1 \end{array} $$

Step-by-Step Solution

Answer

Parametric equations: \(x = 1 - \frac{3}{5}t\), \(y = t\), \(z = -1 + \frac{7}{5}t\).

Step 1: Set Up the System of Plane Equations

We start with the two given plane equations: 1. \(x + 2y - z = 2\)2. \(3x - y + 2z = 1\)The goal is to express the line of intersection for these two planes.

Step 2: Solve for One Variable

Express one of the variables in terms of the others. Solving the first equation for \(z\):\[z = x + 2y - 2\] Now substitute \(z\) in the second equation.

Step 3: Substitute in Second Equation

Substitute \(z = x + 2y - 2\) into the second equation:\[3x - y + 2(x + 2y - 2) = 1\]Simplify this to:\[3x - y + 2x + 4y - 4 = 1\]\[5x + 3y = 5\]

Step 4: Express a Variable in Terms of the Other

Solve for \(x\) in terms of \(y\) from the equation \(5x + 3y = 5\):\[5x = 5 - 3y\]\[x = 1 - \frac{3}{5}y\]

Step 5: Express the Remaining Variable

Substitute \(x = 1 - \frac{3}{5}y\) back into the expression for \(z\):\[z = (1 - \frac{3}{5}y) + 2y - 2\]Simplify to find \(z\):\[z = 1 - \frac{3}{5}y + 2y - 2\]\[z = -1 + \frac{7}{5}y\]

Step 6: Formulate Parametric Equations

Using the parameter \(t\), substitute \(y = t\). Thus the parametric equations for the line are:\[x = 1 - \frac{3}{5}t\]\[y = t\]\[z = -1 + \frac{7}{5}t\]

Key Concepts

Line of Intersection

A line of intersection is where two planes meet, forming a line in three-dimensional space. When we have two equations representing these planes, we aim to find where they cross paths. This intersection line can be uniquely identified using parametric equations. Parametric equations describe a line in terms of a single parameter, often denoted as \(t\). This approach allows for a simple and precise representation of the line using mathematical functions relating to a parameter rather than coordinates. In our problem, we find a line of intersection by solving the system of plane equations, which shows us the line, weaving through both planes.

Plane Equations

Plane equations in a three-dimensional space are typically represented in the form \(ax + by + cz = d\), where \(a, b, c,\) and \(d\) are constants, and \(x, y, z\) are the variables that denote coordinates in space. Each equation in this format represents a plane, a flat, two-dimensional surface that extends infinitely in three-dimensional space. In our example:

• The first plane equation is \(x + 2y - z = 2\).
• The second plane equation is \(3x - y + 2z = 1\).

These equations lay the groundwork for understanding the spatial dynamics and finding where they intersect each other. By manipulating these equations, we can find key points and lines of intersection.

System of Equations

Systems of equations consist of multiple equations working together. In our context, we have two plane equations, and solving the system means finding a shared solution that satisfies both. By substituting and expressing variables in terms of each other, we can solve for independent parameters that simplify finding intersection lines or common solutions. Here's how the system was branched into a simpler format:

• Rewriting one equation to express one variable.
• Substituting that expression into the other equation.

Through these steps, systems of equations provide the necessary linkage to transition from separate mathematical entities to a cohesive structure that reveals their interactions.

Mathematical Problem Solving

Mathematical problem solving often involves simplifying complex problems by breaking them down into smaller and more manageable tasks. Starting from understanding the core question, like finding where two planes intersect, to systematically apply mathematical tools and techniques is essential. In solving the problem:

• We first identified the need to manipulate the plane equations.
• Then, we methodically expressed variables in terms of each other, gradually simplifying through substitution and solving steps.
• Lastly, we developed parametric equations, offering a structured and clear answer for the line of intersection.

By focusing on each step carefully, employing methodical reasoning, and leveraging our knowledge of equations and geometry, we can unravel complex scenarios into understandable solutions.