An interesting question:

Question

The edges of a rectangular solid are parallel to the coordinate axes, and it has the points (2, 4, 4) and (6, 9, 1) as diagonally opposite vertices. Make a sketch, labeling each vertex with its coordinates, then find (a) the dimensions of the solid and (b) the length of its diagonal.

Answer 1

We could say:

The top and bottom are parallel to the xy-plane.

The sides are parallel to the xz-plane.

The front and back are parallel to the yz-plane.

If we are parallel to the xy-plane our z-coordinate does not change. The point (6, 9, 1) is lower than (2, 4, 4).

All of the points on the top will have coordinates (x, y, 4) - the same z-coordinate as (2, 4, 4).

All of the points on the bottom will have coordinates (x, y, 1) - the same z-coordinate as (6, 9, 1).

If we are parallel to the xz-plane our y-coordinate does not change. The point (6, 9, 1) is further left than (2, 4, 4) - comparing y-coordinates 9 and 4.

All of the points on the left will have coordinates (x, 9, z) - the same y-coordinate as (6, 9, 1).

All of the points on the right will have coordinates (x, 4, z) - the same y-coordinate as (2, 4, 4).

If we are parallel to the yz-plane our x-coordinate does not change. The point (6, 9, 1) is further forward than (2, 4, 4) - comparing x-coordinates 6 and 2.

All of the points on the front will have coordinates (6, y, z) - the same x-coordinate as (6, 9, 1).

All of the points on the back will have coordinates (2, y, z) - the same x-coordinate as (2, 4, 4).

Putting all this together we get

top, front, right (6, 4, 4) - from (x, y, 4), (6, y, z), (x, 4, z)

top, front, left (6, 9, 4) - from (x, y, 4), (6, y, z), (x, 9, z)

top, back, right (2, 4, 4) - from (x, y, 4), (2, y, z), (x, 4, z)

top, back, left (2, 9, 4) - from (x, y, 4), (2, y, z), (x, 9, z)

bottom, front, right (6, 4, 1) - from (x, y, 1), (6, y, z), (x, 4, z)

bottom, front, left (6, 9, 1) - from (x, y, 1), (6, y, z), (x, 9, z)

bottom, back, right (2, 4, 1) - from (x, y, 1), (2, y, z), (x, 4, z)

bottom, back, left (2, 9, 1) - from (x, y, 1), (2, y, z), (x, 9, z)

We can find distance in space by : D = sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}

a) When finding the dimensions of the solid, we can find the difference of two vertices that would give us the height, width, or length. For example, the distance between the top, front, right and the bottom, front, right would simply be 3 - top, front, right (6, 4, 4) - bottom, front, right (6, 4, 1). The only coordinate that is different is the z-coordinate and the difference is 3.

b) the two given points are on opposite vertices of the solid. The segment connecting these two points would be the diagonal. Find the length of this using the distance formula above.