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

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.
$(-1,0,2) ;(4,2,5)$

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Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system represents points in space using ordered sets of numbers, called coordinates. In three dimensions, these are x, y, and z, which denote the point's position along mutually perpendicular axes, providing the framework for analyzing and solving spatial problems.

Rectangular Box Properties

A rectangular box (or cuboid) is a three-dimensional shape where each face is a rectangle and the edges meet at right angles. When its edges are parallel to the coordinate axes, the vertices can be systematically determined by combining the extreme values (minimums and maximums) of the x, y, and z coordinates.

Vertex Determination through Coordinate Combination

Given two opposite vertices of a rectangular box, the other vertices can be found by taking every possible combination of the minimal and maximal coordinate values for x, y, and z. This approach leverages the structure of a cuboid, where each vertex corresponds to one of the unique combinations obtained from the coordinates of the given vertices.

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