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 $(5,6,1)$ and

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

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Key Concepts

Vertex Combination

The concept of vertex combination involves generating the coordinates of the remaining vertices of a rectangular prism by mixing and matching the coordinates of two opposite vertices. Since the faces are parallel to the axes, each coordinate of the vertices on the opposite corner will be one of the two provided coordinate values.

Axis-Aligned Geometry

When a rectangular box has edges parallel to the coordinate axes, it is considered axis-aligned. This property simplifies calculations since each vertex coordinate can be determined by independently choosing the appropriate x, y, and z values from the two given opposite vertices.

Rectangular Prism

A rectangular prism, often referred to as a box, is a three-dimensional figure with six faces, all of which are rectangles. It is characterized by its length, width, and height. Understanding its structure is crucial for solving problems that involve finding its vertices or volume.

Cartesian Coordinate System

The Cartesian coordinate system is a method of specifying points in space using ordered triples (x, y, z). In three-dimensional geometry, this system is essential for plotting points, calculating distances, and describing the positions of objects like lines, planes, and solids.

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. $(5,6,1) ;(3,8,2)$

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