# Intersections of Planes, Points...

I'm in sixth grade and learning geometry. Can someone tell me if I'm correct?

• 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?

• Can you only choose one? A line and plane can intersect either in a point, or in the entire line (if the line is on the plane). Nothing you've written is wrong, it just may not be "the full story." Aug 28, 2015 at 22:24
• @pjs36 What if I said they are to be 'distinct'? Aug 28, 2015 at 22:25
• @timmysolé Give us the question, the whole question, then give us your answer, your whole answer. Then we will tell you if you are correct. Aug 28, 2015 at 22:44
• @Zenohm That's my question up there. Just tell me if I'm right or wrong. Aug 28, 2015 at 22:49
• @timmysolé Yet, when provided with answers to what appears to be your entire question, you bring up the matter of each item being, as you say, 'distinct'. If you're actually asking about distinct items, then the answer changes. Put relevant information in the question, not in the comments. Aug 28, 2015 at 22:56

1. 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.
2. the intersection of a point and a line, is a point only if the point lays on the line. otherwise the intersection is empty.
3. 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.
4. 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:

• How would the answer change if each was distinct. So a distinct point intersects with another distinct point? Aug 28, 2015 at 22:27
• @timmysolé two distinct points do not intersect, therefore you get the empty set. two distinct lines intersect at most a point. two distinct planes intersect in at most a line. Aug 28, 2015 at 22:42
• So is my answer correct if they are not distinct? Aug 28, 2015 at 22:46
• @timmysolé I added some pictures. That should make things clear. Aug 28, 2015 at 23:05

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

1. 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 $y=x+1$.
2. 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.
3. 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".

• Am I right. This is hard for me to understand. How would the answer change if each was distinct. So a distinct point intersects with another distinct point? Aug 28, 2015 at 22:34
• If they are distinct then they do not intersect. They do not intersect in a point; they do not intersect at all! Like I said - they miss. Try reading through what I have written slowly. You replied 90 seconds after my post. Of course you won't understand it in 90 seconds! Aug 28, 2015 at 22:36
• Wait. I'll try reading it. But can you look at my edit? Aug 28, 2015 at 22:38
• @timmysolé "Distinct" means different. The intersection of a distinct something with a distinct something will be nothing. They will not intersect because they are distinct. Your original post made more sense. There is no easy answer to your questions. The true answer is that you need to give all of your things equations and then fine the dimension of the kernel of that system of linear equations. I'm sorry it's complicated, but that's the general answer. It depends on lots and lots of things. Aug 28, 2015 at 22:42
• Will I be fine with my answer posted above then? As per someone taking high school geometry? Aug 28, 2015 at 22:43

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:

• 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.