Problem 60 - Thomas Calculus

Question

Find a formula for the distance from the point \(P(x, y, z)\) to the $$ \text { a. } x y \text { -plane. } \quad \text { b. } y z \text { -plane. } \quad \text { c. } x z $$

Step-by-Step Solution

Answer

a. Distance to xy-plane: \(\lvert z \rvert\). b. Distance to yz-plane: \(\lvert x \rvert\). c. Distance to xz-plane: \(\lvert y \rvert\).

Step 1: Understanding the Problem

We need to find the distance from a point \( P(x, y, z) \) to each of the three coordinate planes: the \(xy\)-plane, the \(yz\)-plane, and the \(xz\)-plane. Each plane has a specific axis which is considered zero on these planes.

Step 2: Distance to the xy-plane

The \(xy\)-plane is defined as all points where \(z = 0\). The distance from \(P(x, y, z)\) to the \(xy\)-plane is simply the absolute value of the \(z\)-coordinate. Therefore, the distance to the \(xy\)-plane is \(\lvert z \rvert\).

Step 3: Distance to the yz-plane

The \(yz\)-plane is defined as all points where \(x = 0\). The distance from \(P(x, y, z)\) to the \(yz\)-plane is the absolute value of the \(x\)-coordinate. Thus, the distance to the \(yz\)-plane is \(\lvert x \rvert\).

Step 4: Distance to the xz-plane

The \(xz\)-plane is defined as all points where \(y = 0\). The distance from \(P(x, y, z)\) to the \(xz\)-plane is the absolute value of the \(y\)-coordinate. Hence, the distance to the \(xz\)-plane is \(\lvert y \rvert\).

Key Concepts

Distance Formula

The distance formula is a crucial tool when it comes to finding the distance between points in space. In a 3D coordinate system, the distance from a point to a plane is a specific application of this formula. When we're talking about planes like the xy-plane, yz-plane, and xz-plane, the task becomes easier because each of these planes sets one coordinate to zero.
To determine the distance from a point \(P(x, y, z)\) to one of the coordinate planes, you only need to consider the coordinate that is set to zero on the plane. For instance:

• Distance to the xy-plane: Since \(z\) is zero on this plane, the distance is simply \(\lvert z \rvert\).
• Distance to the yz-plane: Here, \(x\) is zero, making the distance \(\lvert x \rvert\).
• Distance to the xz-plane: Since \(y\) is zero, the distance equals \(\lvert y \rvert\).

This straightforward method relies on an understanding of how each plane interacts with a 3D coordinate system, providing an easy way to calculate distances without needing complex formulas.

3D Geometry

3D geometry involves understanding coordinates in three dimensions: the x, y, and z axes. This system is essential for visualizing and solving problems in physics, engineering, and mathematics.
Each point in space is identified by a set of three coordinates \((x, y, z)\). These indicate a position relative to the origin, where all axes intersect, usually noted as \((0, 0, 0)\).
In this realm, we encounter objects such as planes, lines, and shapes that extend into depth beyond the familiar 2D space. A coordinate plane, such as the xy-plane, represents a flat surface where one dimension is ignored—here it's the z-dimension.

• The xy-plane is the flat plane where height (z) is nonexistent, represented by \(z = 0\).
• The yz-plane disregards the width dimension, placed where \(x = 0\).
• The xz-plane excludes the depth dimension, occurring at \(y = 0\).

Understanding these fundamental planes helps in visualizing how objects are spread in 3D environments and assists in tackling problems involving spatial calculations.

Planes in Space

Planes in space are limitless flat surfaces that extend infinitely in two dimensions while being set in a 3D context. They can be tricky because you can't see them in entirety. However, understanding their characteristics allows us to navigate and solve complex spatial problems.
In basic 3D space, coordinate planes specifically refer to the xy-plane, yz-plane, and xz-plane. Each of these planes is a cornerstone of the 3D Cartesian coordinate system:

• The xy-plane consists of all points where \(z = 0\). It essentially describes a flat sheet stretching out over the x and y axes.
• The yz-plane includes points with \(x = 0\), extending over the y and z axes.
• The xz-plane comprises points where \(y = 0\), spreading across the x and z axes.

These planes help define the boundaries and locations within three-dimensional space, making them invaluable in applications ranging from computer graphics to physics simulations. Recognizing how to measure distances to these planes is an essential skill in interpreting and managing 3D geometrical information.