How can 3 planes intersect?

Question

How can 3 planes intersect?

Answer 1

They all have 1 point in common,  like the origin in an x, y, z system

Answer 2

They can intersect in :

1)  on line  ( three sheets of your book)

2) one point ( two side walls of your rom and the sealing )

3) three parallel lines ( three sided prism)

Answer 3

I assume you mean "what are the possible sets of points that belong to all three planes at the same time".

Well they can intersect in a point, along a line, be the same plane three times, or not intersect at all. Let's assume that all three planes are distinct. Thus there are only 3 cases:

Consider the three vectors orthogonal to each plane. Call them v1, v2 and v3.

Case 1: one point intersection

if there is no plane such that v1, v2 and v3 simultaneously belong to it, then the intersection is one point. (think of the faces of a cube)

Case2: one line intersection

if the v1, v2 and v3 are all distinct (up to re-scaling) but they all belong to a given plane, then the intersection is a line (think of the pages in a book)

Case 3: no intersection

assume now that v1=v2 (up to re-scaling). Since the two planes P1 and P2 are distinct and have the same orthogonal vector, it follows that they do not intersect (think of the slices of bread in a sandwich). Thus the three planes P1 P2 and P3 do not intersect simultaneously.

Hope this helps.

Answer 4

They can intersect in one point, the example is cartesian coordinates system, OX, OY and OZ planes intersects in (0,0,0)

Answer 5

They will intersect at a point