Problem 4 - University Calculus: Early Transcendentals

Question

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=1, \quad y=0$$

Step-by-Step Solution

Answer

A vertical line parallel to the z-axis at \((1, 0, z)\).

Step 1: Understand the Equations

The equations given are \(x = 1\) and \(y = 0\). These equations describe conditions that points must satisfy in a 3D Cartesian coordinate system. We need to find a geometric description of the set of points meeting these conditions.

Step 2: Analyze \(x = 1\)

The equation \(x = 1\) represents a plane parallel to the \(yz\)-plane in the Cartesian coordinate system, since it restricts any point to have an \(x\)-coordinate of 1, but the \(y\) and \(z\) coordinates can take any value.

Step 3: Analyze \(y = 0\)

The equation \(y = 0\) represents another plane parallel to the \(xz\)-plane. It restricts points to only have a \(y\)-coordinate of 0, but \(x\) and \(z\) coordinates can vary.

Step 4: Combine the Conditions

To find the set of points that satisfy both equations simultaneously, we look for points that have \(x = 1\) and \(y = 0\). The intersection of two planes where \(x = 1\) and \(y = 0\) is a line. Specifically, this is the line where \(x = 1\), \(y = 0\), and \(z\) is free to be any real number.

Step 5: Describe the Geometry

The geometric description of the set of points is a vertical line in space. This line is parallel to the \(z\)-axis and passes through the point \((1, 0, 0)\). Any point on this line will have coordinates in the form \((1, 0, z)\), where \(z\) is any real number.

Key Concepts

Intersection of Planes

Imagine a 3D space where we have two conditions or constraints: one is that every point must lie on a specific plane determined by the equation \(x = 1\), and the other that it must also lie on a separate plane defined by the equation \(y = 0\). The task here is to find the intersection, or the common set of points, between these two planes.

Each plane is like a flat sheet extending infinitely in all directions within its defined constraints. The plane \(x = 1\) is like a vertical sheet parallel to the \(yz\)-plane (since it doesn’t interfere with the \(y\) and \(z\) coordinates), but it is positioned at \(x = 1\). On the other hand, the plane \(y = 0\) is a horizontal sheet that sits flat along the \(xz\)-plane at any \(z\) value, with all points having \(y = 0\).

Where these two planes intersect, they share a common line. In simpler terms, if you think of holding two sheets of paper so they cross each other, the line where they touch each other represents their intersection. This intersection is where both conditions \(x = 1\) and \(y = 0\) are satisfied simultaneously.

Geometric Description

In geometry, finding the intersection of constraints like these results in a new geometric shape or figure which here is a line in three-dimensional space.

When we talk about a geometric description, we're focused on identifying what form is created by the intersection of planes or lines. In this exercise, the conditions \(x = 1\) and \(y = 0\) create a vertical line. This line is distinct because it satisfies both equations: \(x\) is always \(1\) and \(y\) is always \(0\), while the \(z\)-coordinate can be any possible value.

Thus, the geometric description of the set meeting these equations is a straight, unending line parallel to the \(z\)-axis. This line runs through the point \((1, 0, 0)\), and extends infinitely upwards and downwards along the \(z\)-coordinate.

• This configuration gives us a simple straight path through space, which is a crucial idea whenever navigating three-dimensional problems.

Vertical Line in Space

A vertical line in space is a line which runs parallel to the \(z\)-axis of the 3D Cartesian coordinate system. Think of it like a flagpole sticking out of a flat ground – it doesn't curve or tilt but just shoots straight up and down.

In this context, the line we discovered as the intersection of the two planes \(x = 1\) and \(y = 0\) is a perfect example of such a line. The line forms a vertical path through the 3D space, maintaining a constant \(x\) coordinate of \(1\) and a constant \(y\) coordinate of \(0\). The only coordinate allowed to change freely is \(z\), making the line extend infinitely along the \(z\)-axis.

This line is special because:

• It gives us a clear way to visualize intersections in 3D space.
• It helps in understanding how complex 3D systems can be broken down into simpler components by analyzing their constraints or equations.

Understanding vertical lines within the 3D Cartesian coordinate system is foundational in both theoretical and applied mathematics, as they help us see beyond the abstract equations into tangible geometry.