Problem 57 - Thomas Calculus

Question

Find parametrizations for the lines in which the planes in Exercises \(57-60\) intersect. $$ x+y+z=1, \quad x+y=2 $$

Step-by-Step Solution

Answer

The line of intersection is parametrized by \((x, y, z) = (t, 2-t, -1)\).

Step 1: Understand Plane Equations

We are given two planes with equations \(x + y + z = 1\) and \(x + y = 2\). The line of intersection of these two planes will have points that satisfy both equations.

Step 2: Express One Variable in Terms of Another

From the second equation \(x + y = 2\), we can express \(y\) in terms of \(x\): \(y = 2 - x\). We will use this to substitute \(y\) in the first equation.

Step 3: Substitute and Simplify

Substitute \(y = 2 - x\) into the first equation \(x + y + z = 1\). This gives \(x + (2 - x) + z = 1\). Simplify to find \(z + 2 = 1\), resulting in \(z = -1\).

Step 4: Solve for All Variables in Terms of a Parameter

Introduce a parameter \(t\), letting \(x = t\). Then, from step 2, \(y = 2 - t\) and from step 3, \(z = -1\). The line can be parametrized as \((x, y, z) = (t, 2-t, -1)\).

Step 5: Write Parametric Equations

The parametric equations for the line of intersection are: \(x = t\), \(y = 2 - t\), and \(z = -1\).

Key Concepts

Plane Intersection

When two planes meet, their intersection forms a line. To find this line, we need to find points that sit on both planes. For our exercise, we have two plane equations: \(x + y + z = 1\) and \(x + y = 2\).

• The intersection line is made up of points that satisfy both equations at the same time.
• This means we look for values of \(x\), \(y\), and \(z\) that fit into both equations without contradiction.

Since we are dealing with linear equations, the solution to this system will often involve parameters, which then give us a continuous line along the intersection of the two planes.

Line Parametrization

Line parametrization is a method used to express the coordinates of any point on a line using one or more parameters. For our intersection line,

• we start by choosing one parameter, typically denoted as \(t\).
• We then express the coordinates \(x\), \(y\), and \(z\) in terms of \(t\).

From the step-by-step solution:

• \(x\) was chosen as the parameter: \(x = t\).
• From the substitution \(y = 2 - x\), we find \(y = 2 - t\).
• With \(z = -1\), as derived previously, it remains constant.

Consequently, the parametric equations become \(x = t\), \(y = 2 - t\), and \(z = -1\). These equations help us identify any point on the line by simply plugging different values for \(t\). We see how easily these equations can reconstruct the line of intersection.

System of Equations

A system of equations is a collection of equations that are solved together. In this case, we solve the system formed by the two given planes:

• \(x + y + z = 1\)
• \(x + y = 2\)

Solving these equations together allows us to find the exact points where both conditions are met - essentially, those points where the two planes intersect.
The first step in solving such a system is to express some variables in terms of others. Here, \(y = 2 - x\) from the second equation. This can then be substituted into the first equation to solve for \(z\), giving \(z = -1\).
Solving such systems is key to uncovering the nature of intersections in geometry. Each solution represents a point on the line, solidifying the relationship between algebra and geometry.