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First, let me see if I understand your figure correctly. It shows plane A. Points M, V, N, and P are on plane A. There is a line that intersects the plane at point N. Points T and W are points on the line, both being different from point N.
from my understanding, yes
Also, line MP is on the plane.
where is exactly plane NPT?
Think of plane A as being a sheet of paper. On that sheet of paper, there is line MP, so points M and P are on the sheet of paper. Point V is also on the sheet of paper. Now take a pencil and pierce the line with the pencil. Have some of the of pencil stick out the top of the plane and some of the pencil stick out the bottom of the plane. Think of the tip of the pencil as point W and the eraser of the pencil as point T.
Since line MP on the plane and line TW intersect, there is only one plane that contains those two lines.
Every point on lines MP and TW is on the plane that contains points N, P, and T. Points on plane A that are not on line MP are not coplanar with the plane that contains points N, P, and T.
I can imagine the pencil sticking out and where all the points lie.. in reality, but after that I'm lost. From my understanding, a plane is like a shape right? (I.E being the paper). What is plane NPT resemble in real life?
i think i understand it now, but would T and W be coplaner with points n p t? @mathstudent55
yes T and W would be coplanar with N, P, and T because they are on line TW.
A plane is a flat surface. That's why I used a sheet of paper to represent the plane.
@mathstudent55, so is the "air" (i assume thats the plane the pencil eraser and bottom is sticking out) is considered, a plane? Apparnetly, there's two planes, plane a and plane NPT.
Perhaps this drawing shows you our situation more clearly. Plane A and the plane containing point N, P, and T extend forever, just like a line extends in both directions forever. |dw:1440093983261:dw|
In the figure above, you see clearly that points M, N, P, W, and T are all on the plane that contains points N, P, and T. Also you see that point V, which is on plane A, is not coplanar with points N, P, and T.
Yeah. i can see that fully once u draw it like that, but i have problems understanind where did the other shape come form? I know its imaginary, but i don't understand how did you come up with that? Is it because point N P T (n goes all the way down) P goes all the way to the right and T goes all the way to the top, and thus forms a shape for the "second plane"?
A plane is infinite in length and width, just like a line has an infinite length. In my figures, I used rectangles or parallelograms to represent planes, but remember they are just representations. When we draw a line and want to show the line goes on forever in both directions, we put arrowheads at both ends of the line segment to show the length of a line is infinite. It's not practical to put arrowheads at the edges of a rectangle or parallelogram, so it looks like a polygon of a certain size, but it is not. Once again, a plane is a flat surface that has length and width, but both the length and width are infinite.
If you have a single line, there are many planes that contain that line. |dw:1440168020188:dw|
In the figure above, you see line m. There is an infinite number of planes that contain line m. Planes A and B are two such planes. If you rotate plane A about line m, there is an infinite number of angles that you can have the plane at, so there is an infinite number of planes through line m.
If you have two different lines that intersect at one point, then they define one single plane.
The figure above shows two different lines that intersect at a point. There is only one plane that contains those two lines. That plane is called plane C. When a plane contains a line, that means all points of the line are in the plane. In our case, all points of lines a and b are contained in plane C. Remember, just like the lines extend indefinitely, the planes also do. A plane is an infinitely sized flat surface with no thickness. A line is an infinitely sized set of points with only length but not width or height. A line only has length, and the length is infinite.
Any two points define a single line. If you are given a single point in space, there is an infinite number of lines that contain that point. If you are given two points in space, there is a single line that contains those points.
Go over the info above. Then you'll understand the following. Now let's get back to our problem. Any three non-collinear points in space define a single plane. You are given points N, P, and T. You can draw a line between any two of those points, but you can't draw a line through all three. That means the points are non-collinear. Since points N, P , and T are non-collinear, there is a single plane that contains those three points. That is the plane that looks vertical in the red drawing.
If you look in the red figure, you see that point V is on plane A, but not on the vertical plane. Point V is non-coplanar with points N, P, and T.