Main content

## Geometry (all content)

### Course: Geometry (all content) > Unit 1

Lesson 4: Points, lines, & planes# Specifying planes in three dimensions

In a three-dimensional space, a plane can be defined by three points it contains, as long as those points are not on the same line. Learn more about it in this video. Created by Sal Khan.

## Want to join the conversation?

- What does collinear mean?(162 votes)
- Hi Pranav,

Collinear points are points that lie on the same line. If you only have two points, they will always be collinear because it is possible to draw a line between any two points. If you have three or more points, then, only if you can draw a single line between all of your points would they be considered collinear.

Hope that helps!(315 votes)

- I though a plane was two dimensional, if I am wrong can you please explain?(58 votes)
- Planes
*are*two dimensional, but they can exist in three dimensional space.(122 votes)

- I don't understand what names a plane and why you need 3 points(15 votes)
- What is the smallest number of legs a stool can have and still be a free standing stool?

If it has one leg it will fall over... same with two.

If it has three legs it will stand, but only if those three legs are not on the same line... the ends of those three (non-collinear) feet define a plane.

If the stool has four legs (non-collinear) it will stand, but if one of the feet is out of alignment it will wobble... it wobbles between two sets of three legs each... each defines a different plane.

If I remember correctly you can identify a plane with a single capital letter, or any three non-collinear points in that plane... so if plane M contains points a, b and c it could also be called plane abc(164 votes)

- What do collinear and coplanar mean?(19 votes)
- Collinear means "lying on the same line". Two or more points are collinear, if there is one line, that connects all of them (e.g. the points A, B, C, D are collinear if there is a line all of them are on). This means, that if you look at just two points, they are automatically collinear, as you could draw a line that connects them.

Coplanar means "lying on the same plane". Points are coplanar, if they are all on the same plane, which is a two- dimensional surface. Any three points are coplanar (i.e there is some plane all three of them lie on), but with more than three points, there is the possibility that they are not coplanar.(48 votes)

- At2:23he says collinear what does that mean?(9 votes)
- Let's break the word collinear down:

co- : prefix meaning to share. For example, a coworker is someone who shares your work place.

linear: related to a line.

If we put this together, collinear would mean something that shares a line. Or, points that lie on the same line.(40 votes)

- I'm slightly confused on the difference between the 1st, 2nd, and 3rd dimensions. i understand that they each identify how an object occupies space and how it can move in said space (ie; 1st can't move at all, 2nd can only move back and forth or up and down, 3rd can move forwards, backwards, up down, back and forth) but i don't get how i would use this or how it would work in higher powers such as the 4th or 5th and how we have come to understand we live in a universe of dimensions.(4 votes)
- Be careful with what you said. A object in 1-dimensional space can move in exactly one direction. So really it's proper to say:

0D: I can't move anywhere.

1D: I can move in one direction.

2D: I can move in any combination of two directions.

3D: I can move in any combination of three directions.

For higher dimensions, we can't visually see it, but we can certainly understand the concept. For instance, an example of a 4D space would be the world we live in and the dimension of time. We can't see time, but we know that it is independent of the other three dimensions. Hope this helps. Good luck.(8 votes)

- why don't they show us what "coplanar" points in this video. If anyone saw it please tell ,and please explain it to me(3 votes)
- I did not see "coplanar" within this video, but coplanar refers to points that lie on the same
**axis**or**plane**as they keep mentioning. For example, if points A, B and C lie on the X axis, then they are coplanar. If, for example, line GF were represented diagonally, with an interception at point (0,0), and points DEF lie on line GF, then they would all lie on the same**axis**, making them**coplanar**.(5 votes)

- I am still confused about what a plane is. Is it any shape?(3 votes)
- A plane is basically a 2D line. It extends infinitely in 2 axis, instead of one like a line.(1 vote)

- If I have two lines with the exact same coordinates, are they parallel or intersecting?

I am asking that if it looks like there is only one line on a plane, but there are actually two lines and are "lined" :) up on top of each other, is it parallel or intersecting?(2 votes)- They are coincident... they might be considered parallel or intersecting depending on the nature of the question.

Parallel lines typically have no points in common while intersecting lines have one point in common... coincident lines have all points in common(4 votes)

- can we specify the plane by saying ADJ? cause in all the examples, D was left out(3 votes)

## Video transcript

We've already been exposed
to points and lines. Now let's think about planes. And you can view planes as
really a flat surface that exists in three dimensions, that
goes off in every direction. So for example, if I have
a flat surface like this, and it's not curved,
and it just keeps going on and on and
on in every direction. Now the question is, how
do you specify a plane? Well, you might say,
well, let's see. Let's think about
it a little bit. Could I specify a plane with
a one point, right over here? Let's call that point,
A. Would that, alone, be able to specify a plane? Well, there's an
infinite number of planes that could go
through that point. I could have a plane that
goes like this, where that point, A,
sits on that plane. I could have a plane like that. Or, I could have
a plane like this. I could have a plane
like this where point A sits on it, as well. So I could have a
plane like that. And I could just keep
rotating around A. So one point by
itself does not seem to be sufficient
to define a plane. Well, what about two points? Let's say I had a point,
B, right over here. Well, notice the way I
drew this, point A and B, they would define a line. For example, they would define
this line right over here. So they would define,
they could define, this line right over here. But both of these points and
in fact, this entire line, exists on both of these
planes that I just drew. And I could keep
rotating these planes. I could have a plane
that looks like this. I could have a plane
that looks like this, that both of these
points actually sit on. I'm essentially just
rotating around this line that is defined by
both of these points. So two points does not
seem to be sufficient. Let's try three. So there's no way
that I could put-- Well, let's be careful here. So I could put a third
point right over here, point C. And C
sits on that line, and C sits on all
of these planes. So it doesn't seem like
just a random third point is sufficient to define, to pick
out any one of these planes. But what if we
make the constraint that the three points are
not all on the same line. Obviously, two points
will always define a line. But what if the three
points are not collinear. So instead of
picking C as a point, what if we pick--
Is there any way to pick a point, D, that
is not on this line, that is on more than one
of these planes? We'll, no. If I say, well, let's see,
the point D-- Let's say point D is right over here. So it sits on this
plane right over here, one of the first
ones that I drew. So point D sits on that plane. Between point D, A, and
B, there's only one plane that all three of
those points sit on. So a plane is defined by
three non-colinear points. So D, A, and B, you see, do
not sit on the same line. A and B can sit
on the same line. D and A can sit
on the same line. D and B can sit
on the same line. But A, B, and D does not sit
on-- They are non-colinear. So for example, right
over here in this diagram, we have a plane. This plane is labeled, S. But another way that we
can specify plane S is we could say, plane--
And we just have to find three non-collinear
points on that plane. So we could call this plane AJB. We could call it plane JBW. We could call it plane-- and I
could keep going-- plane WJA. But I could not specify
this plane, uniquely, by saying plane ABW. And the reason why
I can't do this is because ABW are
all on the same line. And this line sits on an
infinite number of planes. I could keep rotating around the
line, just as we did over here. It does not specify
only one plane.