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.
$(-2,-3,0) ; \quad(-6,7,1)$

Key Concepts

Vertex Construction

This concept involves determining all other vertices of an axis-aligned rectangular prism by combining the different coordinate values from two given opposite vertices. Since each vertex is made up of one coordinate from each axis, the remaining vertices can be generated by alternating the coordinate values appropriately.

Opposite Vertices

In a rectangular box, opposite vertices are any two vertices that lie at the greatest distance apart from each other. These vertices define the bounds of the box and are key to determining the coordinates of the remaining vertices through combinations of their coordinate values.

Rectangular Prism

A rectangular prism, or box, is a three-dimensional figure with six rectangular faces. In the context of coordinate geometry, its vertices can be determined by considering the combinations of the minimum and maximum coordinates along each axis.

Cartesian Coordinates in 3D

This concept involves representing points in three-dimensional space using three numerical coordinates, which correspond to positions along the x-, y-, and z-axes. It is fundamental in geometry and physics for describing spatial relationships and positions.

Axis Alignment

When a geometric object’s edges are parallel to the coordinate axes, the object is said to be axis-aligned. This property simplifies the determination of its vertices, as each coordinate of the vertices will be either the minimum or maximum value along that axis.

Please give Ace some feedback