Chapter 9.6, Problem 21AYU

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.

$\left(0,0,0\right);\left(2,1,3\right)$

To determine

To find: List the coordinates of the other six vertices of the box.

Answer to Problem 21AYU

Solution:

$\left(2,0,0\right)$, $\left(2,\text{ 1},0\right)$, $\left(0,\text{ 1},0\right)$, $\left(2,\text{ 0},\text{ 3}\right)$, $\left(0,\text{ 0},\text{ 3}\right)$, $\left(0,\text{ 1},\text{ 3}\right)$

Explanation of Solution

Given:

$\left(0,\text{ 0},\text{ 0}\right)$; $\left(2,\text{ 1},\text{ 3}\right)$

Calculation:

It's a rectangular prism with all edges parallel to coordinate axes. So the two endpoints of each edge must have two coordinates the same, and the third coordinate different.

For all eight vertices, there will be only two values for each coordinate. We're given those values, so all we need is to figure out the other six combinations. The easiest way to generate them is to combine each pair of coordinates for each of the given points with the other possible value for the other one.

So we generate coordinates for three vertices using pairs from $\left(0,\text{ 0},\text{ 0}\right)$:

$\left(2,\text{ 0},\text{ 0}\right)$ [changing the first one].

$\left(0,\text{ 1},\text{ 0}\right)$ [changing the middle one].

$\left(0,\text{ 0},\text{ 3}\right)$ [changing the last one].

We get the other three by using a pair from $\left(2,\text{ 1},\text{ 3}\right)$ for each:

$\left(0,\text{ 1},\text{ 3}\right)$ [changing the first one].

$\left(2,\text{ 0},\text{ 3}\right)$ [changing the middle one].

$\left(2,\text{ 1},\text{ 0}\right)$ [changing the last one].

Therefore,

$\left(2,\text{ 0},\text{ 0}\right)$, $\left(2,\text{ 1},\text{ 0}\right)$, $\left(0,\text{ 1},\text{ 0}\right)$, $\left(2,\text{ 0},\text{ 3}\right)$, $\left(0,\text{ 0},\text{ 3}\right)$, $\left(0,\text{ 1},\text{ 3}\right)$