Problem 64 - University Calculus: Early Transcendentals

Question

Find parametrizations for the lines in which the planes. $$5 x-2 y=11, \quad 4 y-5 z=-17$$

Step-by-Step Solution

Answer

Parametrization: \(x = \frac{11 + 2t}{5}, y = t, z = \frac{4t + 17}{5}\)

Step 1: Express Variables

There are three variables: \(x\), \(y\), and \(z\). Our first plane is defined by the equation \(5x - 2y = 11\) and the second plane is defined by \(4y - 5z = -17\). We will express two of these variables in terms of parameters. Typically, it is convenient to let one of the parameters be \(t\).

Step 2: Solve for \(x\) and \(z\)

To express \(x\) in terms of \(t\), let \(y = t\). Substitute \(y = t\) into the first plane equation: \(5x - 2t = 11\). Solve for \(x\):\[5x = 11 + 2t \]\[x = \frac{11 + 2t}{5}\]Now solve for \(z\) in terms of \(t\) using the second plane equation: substituting \(y = t\) into \(4t - 5z = -17\):\[4t = 5z - 17\]\[z = \frac{4t + 17}{5}\].

Step 3: Write Full Parametrization

With \(y = t\), \(x = \frac{11 + 2t}{5}\), and \(z = \frac{4t + 17}{5}\), we now have the parametric equations:\[x = \frac{11 + 2t}{5}, y = t, \z = \frac{4t + 17}{5}\]

Key Concepts

Plane Intersection

The intersection of two planes results in either:

• A line, if the planes are neither parallel nor coincident.
• A plane, if the planes coincide.
• No intersection, if the planes are parallel but not coincident.

In this exercise, we are interested in when two planes intersect in a line. Imagine two sheets of paper meeting at a common edge. This line of intersection is a new geometric entity, having its own properties.

To find this intersecting line, we typically set up a system of equations using the equations of the planes. In our example, these equations are: \(5x - 2y = 11\) and \(4y - 5z = -17\). We solve this system to find a relationship between the variables that defines the intersecting line. The key to identifying the intersection as a line is that both plane equations can be solved simultaneously without contradicting each other's solutions.

System of Equations

The idea of solving a system of equations involves finding values for the variables that satisfy all equations simultaneously. When two planes intersect, the equations represent constraints on the variables.

In our example, the system of equations is:
1. \(5x - 2y = 11\)
2. \(4y - 5z = -17\)

Step-by-step, we start by isolating one of the variables. Here, we chose \(y\) to be a parameter \(t\), simplifying the system. Then the equations become linear with respect to \(x\) and \(z\). This substitution reduces the complexity of the equations, allowing us to express \(x\) and \(z\) in terms of \(t\).

Solving these equations yields a consistent solution for all three variables. This solution represents the parametric form of the line at the intersection of the planes.

Parametrization of Lines

Parametrization is a technique for expressing a line or curve using a parameter. This process gives us a set of equations that define all the points on a line as a function of a common parameter, often \(t\).

Imagine a slider moving along a line, with each position on the slider corresponding to a particular value of \(t\). This is effectively what parametrization accomplishes. For our problem, with \(y = t\), we've developed the following parametric equations:

• \(x = \frac{11 + 2t}{5}\)
• \(y = t\)
• \(z = \frac{4t + 17}{5}\)

These equations describe all points on the line of intersection by varying \(t\).

Parametrization is crucial for simplifying complex geometric shapes into manageable mathematical expressions, making it easier to analyze, compute, and graph them.