In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordi

Step 1:u000AIdentify the given vertices of the rectangular box. The vertices given are $(0,0,0)$ and

In Problems 21-26, opposite vertices of a rectangular box...

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Key Concepts

Cartesian Coordinate System

This concept involves representing points in space using ordered triples of numbers, which define positions along mutually perpendicular axes. It provides the framework for analyzing geometric objects in three dimensions by assigning numerical values to their positions.

Rectangular Box

A rectangular box, or rectangular prism, is a three-dimensional figure with six rectangular faces. Its edges are perpendicular to each other and meet at right angles, making calculations and reasoning about its dimensions straightforward using basic arithmetic with coordinates.

Vertices Determination by Coordinate Combinations

The vertices of a box aligned with the coordinate axes can be determined by taking every possible combination of the minimum and maximum coordinate values along each axis. In the context of a problem where opposite vertices are given, this process involves mixing the x, y, and z coordinates from both vertices to find all remaining corners.

Axis Alignment

When a solid is aligned with the coordinate axes, its faces are parallel to the coordinate planes. This orientation simplifies the computation and enumeration of vertices, as each edge lies along a single coordinate axis, allowing for a direct substitution of coordinate values.

In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. $(0,0,0) ;(2,1,3)$

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