Question

Opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are (-5, -6, 3) and (-9, 10, 4) . List the coordinates of the other six vertices of the box.

Answer 1

Answer:

C=(-9,-6,-3)

D=(-9,10,-3)

E=(-5,10,-3)

F=(-5,10,4)

G=(-5,-6,4)

H=(-9,-6,4)

Step-by-step explanation:

Its always convenient to name the two points; let's name (-5,-6,3) and (-9,10,4) as A and B respectively.

Also, we can say that these points are Position vectors from the origin!

$\overrightarrow{OA}=\begin{bmatrix}-5\\6\\3\end{bmatrix},\quad\overrightarrow{OB}=\begin{bmatrix}-9\\10\\4\end{bmatrix}$

So in order to find the dimensions of the box, we need to find the distance from OA to OB (aka relative distance $\overrightarrow{AB}$)

$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$

$\overrightarrow{AB}=\begin{bmatrix}-9\\10\\4\end{bmatrix}-\begin{bmatrix}-5\\-6\\3\end{bmatrix}$

$\overrightarrow{AB}=\begin{bmatrix}-4\\16\\1\end{bmatrix}$

Now, why did we do this?

• the vector AB shows the 'direction to go' from the point A to the point B. Now think of point A as it is on the the origin (0,0,0) and if we move -4 steps in the x direction, 16 steps to the y-direction and 1 step to the z-direction, you'll reach the point B. (the directions can be taken in any order)

• Another cool thing here is that at each individual direction, you're also crossing other vertices of the box.

• To find the actual coordinates: just add the coordinates of A to the relative vector positions (don't worry if this is confusing, we'll do this below)

So at the point A, the relative distance from A is (0,0,0)

if you add the coordinates of the point A (-5,-6,3) to (0,0,0) you'll get its original position!

a) If we move -4 steps in the direction of x. The relative distance from A will be: (-4,0,0), this is a vertex of the box.

it's original position will be (-4,0,0) + (-5,-6,3) = (-9,-6,-3), let's call it vertexC.

b) If we move 16 steps more in the direction of y. The relative distance from A will be: (-4,16,0) this is a vertex of the box.

it's original position will be (-4,16,0) + (-5,-6,3) = (-9,10,-3), let's call it vertexD.

If we move 1 step more in the direction of z. The relative distance from A will be: (-4,4,1) this is a vertex of the box.

it's original position will be (-4,16,1) + (-5,-6,3) = (-9,10,4) we're back to our point B.

we have found the coordinates of 2 more vertices of the box in total (we need to find 4 more!). Previously, we first took the x-direction then y and z. now we're gonna take the y-direction first, its a different route, but will also take us to B. This will help us meet the other edges of the box

let's go back to our relative position of A (our starting point). (0,0,0)

c) If we 16 steps in the direction of y. The relative distance from A will be: (0,16,0) this is a vertex of the box.

it's original position will be (0,16,0) + (-5,-6,3) = (-5,10,-3), let's call it vertexE.

d) If we move 1 step more in the direction of z. The relative distance from A will be: (0,16,1) this is a vertex of the box.

it's original position will be (0,16,1) + (-5,-6,3) = (-5,10,4), let's call it vertexF.

if we move in the direction of x, we'll be at B. So we're gonna go back and take another different route. This time, we'll start from the z-direction.

let's go back to our relative position of A (our starting point). (0,0,0)

e) If we move 1 step in the direction of z. The relative distance from A will be: (0,0,1) this is a vertex of the box.

it's original position will be (0,0,1) + (-5,-6,3) = (-5,-6,4), let's call it vertexG.

f) If we move -4 steps in the direction of x. The relative distance from A will be: (-4,0,1), this is a vertex of the box.

it's original position will be (-4,0,1) + (-5,-6,3) = (-9,-6,4), let's call it vertexH.

and we have found all the coordinates of the vertices of the box.!