Intersections of Planes, Points... I'm in sixth grade and learning geometry. Can someone tell me if I'm correct?


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*The intersection of a point and a point is a point.

*The intersection of a point and a line is a point.

*The intersection of a point and a plane is a point.

*The intersection of a line and a point is a point.

*The intersection of a line and a line is a point.

*The intersection of a line and a plane is a point.

*The intersection of a plane and a point is a point.

*The intersection of a plane and a line is a point.

*The intersection of a plane and a plane is a line.


Something seems wrong to me here. Can someone check this.
Also is it the same when asking for the intersection of a plane vs point, and point vs plane or something along those terms as seen in the statements above?
 A: *

*The intersection of a point and a point is a point if they are the same. If they are two different points, then the intersection is empty.

*the intersection of a point and a line, is a point only if the point lays on the line. otherwise the intersection is empty.

*The intersection of a line and a line, may be a point (if they cross each other but are different from each other). But it can also be a line, if both lines are the same (on top of each other so to speak). And it may be empty if both lines don't touch each other.

*The intersection of a line and a plane may be a point. But it can also be a line, if the line lies in the plane. or it may be empty if the line and the plane don't touch.


Maybe a picture will help:


A: There are three important things that you need to think about:


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*Things can miss each other, i.e. be "disjoint". For example, what is the intersection of the points $(0,0)$ and $(1,1)$? What about the lines $y=x$ and $y=x+1$.

*If they do intersect, what if they overlap? In two dimensions: the line $y=x$ and the line $x-y=0$ intersect along a line. In three dimensions: the planes $x=0$ and $y=0$ meet along a line, but the planes $x=0$ and $2x=0$ meet in a plane.

*If the objects do intersect, then the number of dimensions you're working in is important. For example: are you in the plane, or in 3d, or 4d?


In three dimensions, intersecting planes usually meet along a line. In four dimensions they usually meet at a point. 
This can all get quite complicated. In three dimensions, a plane is given by one linear equation, e.g. $x+2y+3z=1$. Solving that one equation imposes one condition and makes you drop down from all of 3d to a 2d plane. To intersect two planes you need to solve two equations at once. So you impose two conditions, say $x=0$ and $y=0$. That drops you down from 3d to 1d. You have imposed two conditions, or used two degrees of freedom, so you drop down by two dimensions.
In four dimensions, a plane is given by two equations. A plane is 2d and so solving two equations drops you down from 4d to a 2d plane. Intersecting two planes in 4d means you have to solve four equations at once. So you use up four degrees of freedom and drop down from 4d to 0d, i.e. a point.
That is what usually happens. Sometimes things are more complicated. Again in 4d. What is one plane has equations $x=0$ and $y=0$, and the other has equations $y=0$ and $z=0$? These aren't really four equations. The equation $y=0$ appears twice. There are only three "independent" equations. So you have only imposed three conditions, you have only used three degrees of freedom. You will only drop down from 4d to 1d. This time, the planes intersect to give a line.
Take a look at this Wikipedia article on "Systems of Linear Equations".
A: Let's look at each in turn,


*

*The intersection of a point and a point is a point.
Two intersecting points can only result in a point, this is correct.

*The intersection of a point and a line is a point.
A point can only intersect with a line at one point on the line, this is correct.

*The intersection of a point and a plane is a point.
A point can only intersect with a plane at one point on said plane, this is correct.

*The intersection of a line and a point is a point.
See point + line.

*The intersection of a line and a line is a point.
This is true except when the lines lie on each other, in which case the intersection would be a line.

*The intersection of a line and a plane is a point.
This is true except when the line lies on the plane, in which case the intersection would be a line.

*The intersection of a plane and a point is a point.
See point + plane.

*The intersection of a plane and a line is a point.
See line + plane.

*The intersection of a plane and a plane is a line.
This is true except when the planes lie on each other, in which case the intersection would be a plane.

What you can take away from this is that:


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*These rules for intersections, like basic addition, are commutative.

*The least complex item in an intersection will always be the limiting factor, no two intersecting items can ever result in anything more complex than the least complex item.

*If you ask a question here, expect that some answers will exceed your grade level.

*Yes, your statements are correct, but they can be expanded as seen above.
