Maths - distance between parallel planes?

Question

OK, if a plane is defined as passing through three points, with each point defined X,Y,Z in 3D space. And there is a parallel plane passing through another three points, again, with each point defined X,Y,Z in 3D space. How do you work out the shortest distance between the two planes? Note that each point is defined completely independently of each other.

Answer 1

Find the distance between the parallel planes z = x + 2y + 1 and 3x + 6y - 3z = 4.
Solution: A point must be found that lies on one of the planes. When y = x = 0 in the plane z = x + 2y + 1, there exists the point ( 0, 0, 1 ). To find the distance between the planes we can now use the equation for the distance between a point and a plane.

Answer 2

You can do it by breaking the problem down into 2 steps.

Step 1 is to find out the distance between the 2 points 2 dimensionally - in the X and Y axes. Do this by:

Square Root [(x2-x1)²+(y2-y1)²]

From there you can move on to Step 2, which is to find out the distance 3 dimensionally:

Square Root [(Step 1 value)²+(z2-z1)²]

Answer 3

to be brief, we have to find a 3 points to find 2 vectors that are on the plane. Using the cross product we find an orthogonal vector to the plane. using one of the points and the perpendiclar vector, we can fin the distance to the other plane.

Answer 4

I also have a concern with the unique diagram. The shadows seem to be parallel (and the previous answer looks to anticipate this). if so, then the mild source could desire to be a limiteless distance away. wherein case, why are pencils of equivalent length casting diverse length shadows?

Answer 5

sqrt([x2-x1]^2+[y2-y1]^2+[z2-z1]^2)

Answer 6

about 1000cm, i think