Problem 68 - Advanced Engineering Mathematics

Question

In Problems, find parametric equations for the line of intersection of the given planes. $$ \begin{aligned} 2 x-5 y+z &=0 \\ y &=0 \end{aligned} $$

Step-by-Step Solution

Answer

The parametric equations are \( x = t \), \( y = 0 \), and \( z = -2t \).

Step 1: Identify the Intersection

The line of intersection of the planes can be found by solving the system of equations for points that satisfy both planes. We have the equations 1. \( 2x - 5y + z = 0 \) 2. \( y = 0 \)Using equation (2), substitute \( y = 0 \) into equation (1) to get \( 2x + z = 0 \).

Step 2: Parameterize the Variables

Since we have two unknowns and one equation, let's express one variable in terms of a parameter. Let \( x = t \), where \( t \) is a parameter. Substitute \( x = t \) into the equation \( 2x + z = 0 \) which gives us \( 2t + z = 0 \). Solving for \( z \), we get \( z = -2t \).

Step 3: Write the Parametric Equations

Now that we have expressed both \( x \) and \( z \) in terms of \( t \), the parametric equations for the line of intersection are: \[ x = t \] \[ y = 0 \] \[ z = -2t \].

Key Concepts

Line of Intersection

When two planes intersect, they form a line where all points satisfy both plane equations. To find the line of intersection, we essentially need to determine a set of points that meet the criteria of both planes. In three-dimensional space, this line is characterized by a direction determined by the normal vectors of the intersecting planes. Simplifying the original equations accurately helps establish a common solution, leading us to the parametric form of the line. In this situation, solving the original plane equations will yield such a set, marking the intersection.

System of Equations

The task of finding the line of intersection requires solving a system of equations. A system contains multiple equations that share common variables. In our exercise, the equations are: \( 2x - 5y + z = 0 \) and \( y = 0 \). Solving a system means finding values for these variables that simultaneously satisfy all equations involved. Dealing with systems can involve methods such as substitution or elimination. Here, we use substitution, reducing the system to a simpler expression by inserting \( y = 0 \) into the first equation. This step simplifies our search for the line of intersection.

Parametrization

Parametrization involves expressing the coordinates of points on a line or curve in terms of a single parameter. This is useful when dealing with lines of intersection where we eventually want a clear and understandable representation of all points on that line. For example, given a reduced equation from the system, we can set a variable equal to a parameter \( t \). Here, we choose \( x = t \) and solve for other variables accordingly. This method simplifies complex relationships, allowing us to write each coordinate as functions of \( t \), resulting in parametric equations like \( x = t \), \( y = 0 \), and \( z = -2t \). These represent the infinite points along the line.

Planes in 3D Geometry

Planes in three-dimensional geometry are flat, two-dimensional surfaces that extend infinitely in all directions within their plane. Defined by linear equations, a plane is typically represented by an equation of the form \( ax + by + cz = d \). Here, each plane can be visualized as a slab of space cutting through the 3D environment. Finding intersections, such as lines, between two planes is common, demonstrating how they interact. The angle between two intersecting planes, for example, can reveal the nature of their relationship. Understanding planes is crucial for visualizing and solving spatial problems involving lines and surfaces.