Problem 63 - University Calculus: Early Transcendentals

Question

Find parametrizations for the lines in which the planes. $$x-2 y+4 z=2, \quad x+y-2 z=5$$

Step-by-Step Solution

Answer

The parametrization is \( (x, y, z) = (4, 2t + 1, t) \).

Step 1: Write the System of Plane Equations

We start by writing down the given equations of the planes: \[ x - 2y + 4z = 2 \]\[ x + y - 2z = 5 \] Our goal is to find the line of intersection of these two planes.

Step 2: Solve the System for Two Variables

Next, we solve this system for two of the variables in terms of the third, say, solve for \( x \) and \( y \) in terms of \( z \). Subtract the second equation from the first to eliminate \( x \):\[(x - 2y + 4z) - (x + y - 2z) = 2 - 5 \]\[-3y + 6z = -3 \]\[ y = 2z + 1 \]Substitute \( y = 2z + 1 \) into the first equation:\[ x - 2(2z + 1) + 4z = 2 \]\[ x - 4z - 2 + 4z = 2 \]\[ x = 4 \]

Step 3: Find the Parametric Equations

Now that we have \( x = 4 \) and \( y = 2z + 1 \), express \( x \), \( y \), and \( z \) parametrically with parameter \( t = z \). \[ x = 4 \]\[ y = 2t + 1 \]\[ z = t \]

Step 4: Write Parameterized Line of Intersection

The parametric equations for the line of intersection of the two planes are:\[ \begin{align*} x &= 4, \ y &= 2t + 1, \ z &= t\end{align*}\]Thus, the parameterization of the line is \( (x, y, z) = (4, 2t + 1, t) \).

Key Concepts

Parametric Equations

Parametric equations are a powerful way to express the coordinates of the points on a line, curve, or surface using one or more parameters. By introducing parameters instead of directly expressing each variable in terms of the others, we gain flexibility and clarify complex relationships between them.
In three dimensions, as in the case of finding a line of intersection between two planes, we use a single parameter. This is because we are dealing with lines, which are one-dimensional entities.

Here's how parametric equations function in the context of our example:

• Each variable, \( x \), \( y \), and \( z \), is expressed in terms of a parameter \( t \).
• In our solution, \( x \) is a constant (not dependent on \( t \)), meaning it remains at \( 4 \) for any \( t \).
• Both \( y \) and \( z \) change with \( t \); \( y = 2t + 1 \) and \( z = t \).

Using parametric equations simplifies understanding and plotting the line of intersection, as each set of \( (x, y, z) \) can be easily calculated by choosing any value of \( t \). Every point on this line corresponds to a value of \( t \).

System of Linear Equations

A system of linear equations consists of two or more linear equations with common variables that you solve together. When dealing with planes in three-dimensional space, each plane can be described by a linear equation in variables \( x \), \( y \), and \( z \).
To find the line where two planes intersect, we look for simultaneous solutions of their equations. This involves the following:

• Identify two equations representing the planes, such as \( x - 2y + 4z = 2 \) and \( x + y - 2z = 5 \).
• Eliminate one variable to express the others in terms of remaining parameters, simplifying the problem.

As shown in the example, subtracting one equation from the other allows us to eliminate \( x \). This enables solving for \( y \) in terms of \( z \), and then substituting back to find \( x \). Such steps effectively solve for the system and derive parametric equations of the intersecting line.
Solving systems of linear equations this way can also be used in various applications, such as calculating circuit currents in electronics or understanding economic models.

Planes

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. When dealing with equations involving planes, we often use a linear equation of the form \( Ax + By + Cz = D \) to represent a plane in three-dimensional space.

The key characteristics of a plane include:

• It is defined by three non-collinear points or by a point and a normal vector (a perpendicular vector).
• The normal vector to a plane, expressed as \((A, B, C)\), is vital in understanding the plane's orientation in space.

For the interacting planes given by \( x - 2y + 4z = 2 \) and \( x + y - 2z = 5 \), their normal vectors are \((1, -2, 4)\) and \((1, 1, -2)\) respectively.
The intersection of planes occurs along a line, where though infinite in two dimensions, they come together in a single dimension. Finding this line requires systems of equations and parametric formulations to express it fully. Understanding the equations of planes helps in visualizing and calculating how they interact, leading to practical applications in fields like computer graphics and engineering.