Problem 25 - Thomas Calculus

Question

In Exercises \(25-34\) , describe the given set with a single equation or with a pair of equations. The plane perpendicular to the $$\begin{array}{ll}{\text { a. } x \text { -axis at }(3,0,0)} & {\text { b. } y \text { -axis at }(0,-1,0)} \\ {\text { c. }} & {z \text { -axis at }(0,0,-2)}\end{array}$$

Step-by-Step Solution

Answer

a. \(x = 3\); b. \(y = -1\); c. \(z = -2\).

Step 1: Understanding the Plane Perpendicular to the x-axis

When we talk about a plane perpendicular to the x-axis, we mean that the plane is parallel to the yz-plane. A plane parallel to the yz-plane can be represented by the equation \(x = c\), where \(c\) is a constant. Since this plane passes through the point \((3,0,0)\), we have \(x = 3\).

Step 2: Solving for Plane Perpendicular to y-axis

For a plane perpendicular to the y-axis, the plane is parallel to the xz-plane. A plane parallel to the xz-plane is represented by the equation \(y = d\), where \(d\) is a constant. Given that this plane passes through \((0, -1, 0)\), the equation is \(y = -1\).

Step 3: Addressing Plane Perpendicular to z-axis

A plane perpendicular to the z-axis is parallel to the xy-plane. A plane parallel to the xy-plane has the form \(z = e\), where \(e\) is a constant. With the specified plane passing through the point \((0, 0, -2)\), the equation becomes \(z = -2\).

Key Concepts

Equations of Planes

Understanding equations of planes is crucial in coordinate geometry. Planes in a 3D coordinate system can be represented by linear equations. Each plane has an equation that relates the x, y, and z coordinates.

Consider how a plane's orientation affects its equation. A plane parallel to one of the coordinate planes can be described with a simpler equation involving one constant value. For example:

• For a plane parallel to the yz-plane, the equation is of the form \(x = c\), meaning all points share the same x-coordinate.
• For a plane parallel to the xz-plane, the form is \(y = d\), where y is constant.
• A plane parallel to the xy-plane takes the form \(z = e\), with a constant z-coordinate.

Using these equations, we can deduce the spatial orientation of a plane. It simplifies the understanding of how planes position themselves in relation to the 3D axes.

Perpendicular Planes

A perpendicular plane in a 3D coordinate system refers to a plane that intersects an axis at a right angle. This geometric property is significant because it helps define relationships between different planes and lines.

To understand how perpendicular planes are expressed, consider these scenarios:

• A plane perpendicular to the x-axis must be parallel to the yz plane. Therefore, its equation will solely account for the x-coordinate, leaving y and z variable, for instance, \(x = c\).
• Similarly, when a plane is perpendicular to the y-axis, it runs parallel to the xz-plane and is represented by \(y = d\).
• Lastly, a plane perpendicular to the z-axis aligns parallel to the xy-plane and is written as \(z = e\).

These orientations show how the planes are arranged in a 3D system, highlighting their interaction with axis perpendicularly.

3D Coordinate System

The 3D coordinate system is an extension of the 2D plane, adding depth with a third coordinate, z. This system uses three perpendicular axes: x, y, and z, which intersect at a point called the origin \( (0,0,0) \).

Each point in this system is defined by an ordered triplet (x, y, z), representing its position in three-dimensional space. This allows for complex geometric shapes and relationships.

• Coordinate Planes: The system is divided into three coordinate planes: xy-plane, yz-plane, and xz-plane.
• Origin: It's the intersection of these three planes, serving as the system's center point.
• Axes: Each axis facilitates measurement in one dimension: left and right (x-axis), forward and backward (y-axis), and up and down (z-axis).

Understanding this framework is fundamental for solving problems involving spatial geometry, such as finding the distance between points, and angles between intersecting lines or planes.