Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

A rectangular box has its faces parallel to the coordinate planes and has \((2,3,4)\) and \((6,-1,0)\) as the end points of a main diagonal. Sketch the box and find the coordinates of all eight vertices.

Step-by-Step Solution

Answer

The vertices are (2,3,4), (6,3,4), (6,-1,4), (2,-1,4), (2,3,0), (6,3,0), (6,-1,0), (2,-1,0).

Step 1: Understanding Diagonal and Vertices

The endpoints of a diagonal of the box are given as \((2,3,4)\) and \((6,-1,0)\). These points represent opposite corners of the box, and the coordinates differ by the dimensions of the box along each axis.

Step 2: Calculate Lengths of the Sides

The difference in the x-coordinates is \(6 - 2 = 4\). Similarly, the difference in y-coordinates is \(3 - (-1) = 4\), and the difference in z-coordinates is \(4 - 0 = 4\). Hence, the dimensions of the box are 4 units along each axis.

Step 3: Identify the Faces Parallel to Coordinate Planes

The faces of the box are rectangular figures parallel to the coordinate planes. We use these face planes to identify all vertices.

Step 4: List All Vertices

From the endpoints \((2, 3, 4)\) and \((6, -1, 0)\), we calculate the remaining vertices of the cuboid as follows:- \((2, 3, 4)\), \((6, 3, 4)\), \((6, -1, 4)\), \((2, -1, 4)\)- \((2, 3, 0)\), \((6, 3, 0)\), \((6, -1, 0)\), \((2, -1, 0)\)These coordinates form the vertices of the box when combined correctly.

Key Concepts

Understanding Coordinate Planes

In 3D geometry, coordinate planes play a vital role. The three main coordinate planes are:

• The XY-plane: This plane is parallel to the x-axis and y-axis. It is "flat" and lies horizontally in a 3D space.
• The XZ-plane: This one is parallel to the x-axis and z-axis, combining horizontal and vertical orientations.
• The YZ-plane: This plane is defined by the y-axis and z-axis, also combining horizontal and vertical orientations.

Each plane divides 3D space and is critical when dealing with problems like a rectangular box where its faces are parallel to these planes. These planes help us visualize where the box sits in the space and understand how its sides align with different axes.
The rectangular box in this exercise has its faces aligned or parallel with these planes, which simplifies finding all the vertices.

Exploring Diagonal Vertices

Diagonal vertices are two points that are situated opposite to each other on a rectangular box. They form a line that passes through the center of the box and cuts it into two equal halves. In our exercise, we have the diagonal endpoints as i.e., (2, 3, 4) and (6, -1, 0). These points are crucial because they provide information about - The box's length in each direction - The potential coordinates of the eight vertices
The difference between the corresponding coordinates of these diagonal vertices gives the dimensions of the box: x-length = 6 - 2 = 4 y-length = 3 - (-1) = 4 z-length = 4 - 0 = 4 This means the box stretches 4 units across each axis. Finding the length helps sketch and understand the scale of the box thoroughly.

Basics of 3D Geometry

The realm of 3D geometry extends through three dimensions: length, width, and height. In 2D, we work on just two axes - x and y. But, in 3D geometry, we introduce an additional z-axis, adding depth. From problems like the rectangular box, we can learn:

• How to visualize objects in three dimensions.
• Understand spatial properties like volume and surface area - although not required here, these concepts are often explored.

In sketching or imagining a box, remember each dimension corresponds to an axis. In our example from the vertices:

• x-axis length holds between points like i.e. 2 and 6.
• y-axis stretches between positions like i.e. 3 and -1.
• z-axis flows from i.e. 4 to 0, establishing the box's height.

All vertices occur at intersections along these dimensions, contributing to a full 3D shape.

Navigating the Coordinate System

A coordinate system helps us to pinpoint locations in a multidi-mensional space using numerical coordinates. In our 3D system:
The point is determined by three numbers - (x, y, z). Each number refers to a distance along an axis:

• x-axis: Usually represents horizontal placement.
• y-axis: Typically indicates horizontal but orthogonal (perpendicular) to x.
• z-axis: Adds depth, moving the point above or below a plane formed by x and y.

For our specific exercise, knowing each axis helps us calculate exact positions for vertices apart from given diagonal vertices. Starting with (2, 3, 4) and (6, -1, 0), we derived other points like (6, 3, 4), extending along the x-axis, or (2, -1, 0) changing y and z values. This systematic approach helps comprehend spatial relationships and object formation in three dimensions.