### Video Transcript

In this video, we will learn how to
identify and model points, lines, and planes in space. Weβll also consider their
properties, along with how they interact with one another. So letβs begin by understanding
exactly how we can define these three objects.

In geometry, a point is a location
in space. It has neither shape nor size. However, when weβre working with
the concept of a point, we have to represent it somehow. And so we represent it with a
point, and generally it is given a letter as a name. We can then go further and say that
we can take any two points in space. We can join these two points with
exactly one straight line between these points. This allows us to define a line as
a connected set of points that extends infinitely in two directions.

We can define this line that joins
π΄ and π΅ in a number of ways, for example, line π΄π΅ with a double-ended arrow over
the letters, line π΅π΄, or using the words βline π΄π΅β or βline π΅π΄,β or even by
defining the line with a single letter, such as line πΏ. Note that the distance between any
two points on a line is called a line segment. This then leads us to the third
important definition for this video, planes.

A plane is a two-dimensional
surface that extends infinitely in all directions. We usually model a plane like
this. And the defining property of a
plane also helps us with naming planes. There exists exactly one plane
through any three noncollinear points. That means that if points π΄, π΅,
and πΆ lie on a plane, we can define the plane as plane π΄π΅πΆ or even plane π΅π΄πΆ
or plane πΆπ΅π΄. Or just like we did with a line, we
can define the plane with a single letter, for example, plane πΎ.

Weβll now see an example where we
identify and define the planes which pass through two given points.

Find three planes that pass through
both of the points π΄ and π΅.

Letβs observe that points π΄ and π΅
lie on the base of this diagram. The planes that pass through both
points π΄ and π΅ will be the planes that pass through the line π΄π΅. We should recall that there exists
exactly one plane through any three noncollinear points. Notice how we use the word
βnoncollinear.β If the three points were collinear,
they would lie on a line. And there would be an infinite
number of planes passing through three collinear points.

So, in order to find a plane
passing through π΄ and π΅, we need to find another point which doesnβt lie on this
line, which could define a plane. Letβs take point πΆ and visualize
the plane that would pass through π΄, π΅, and πΆ. It would look something like
this. And so one of the planes we are
looking for is actually one which contains one of the faces of this prism.

One way in which we can define a
plane is by using three points that we know lie on the plane. So we could call this plane the
plane π΄π΅πΆ or plane π΄πΆπ΅. However, the plane π΄π΅π· would
also work.

Now letβs see if we can find
another plane that passes through π΄ and π΅. We can do this by finding another
point. So letβs use the point π΅
prime. Since we know that π΅ prime is not
collinear with π΄ and π΅, then we can create another plane. We can define this plane as the
plane π΄π΅π΅ prime. We know that we are looking for
three planes passing through π΄ and π΅. So letβs see if we can find the
third plane.

Now, it might be tempting to think
that as we found a plane on the base of the prism and one on the side of the prism
that perhaps there is another plane passing through π΄ and π΅ on one of the other
sides of the prism. Perhaps there might be one on the
back of the prism. However, if we visualize a plane
here, it contains the points π΄, π΄ prime, π·, and π· prime, but it does not contain
the point π΅. So that wouldnβt work.

In fact, the third plane containing
points π΄ and π΅ will look something like this. This plane does contain the points
π΄ and π΅ and πΆ prime and π· prime. So we could define it as the plane
π΄π΅πΆ prime, although it would be equally valid to refer to it as plane π΄π΅π·
prime. Therefore, we can give the answer
that the three planes passing through the points π΄ and π΅ are the planes π΄π΅πΆ,
π΄π΅π΅ prime, and π΄π΅πΆ prime.

So far, we have covered exactly
what we mean by points, lines, and planes in space. Now, letβs think about the way that
these interact with each other in space.

When it comes to two lines in
space, there are three possible relationships that these lines can have, which can
be determined by whether the lines intersect or do not intersect.

Firstly, letβs think about two
lines which are coplanar, which means that they lie on the same plane. Here are two lines which lie on the
same plane, and they are intersecting. Furthermore, if they intersect at
right angles, we could say that the lines are intersecting orthogonally. We can also have two lines which
are coplanar and are nonintersecting, in which case these lines would have to be
parallel. They will never meet.

Now, weβre used to having two types
of relationships between lines: those that intersect and those which donβt. However, in space, there is another
option. These types of lines will be
noncoplanar and nonintersecting. They can be modeled like this. These two lines, β and π, are
called skew lines. They are noncoplanar lines that
donβt intersect because line β occurs in a vertically higher plane than line π. They also cannot be said to be
parallel, because at some points line π is closer to line β than at other
points. Notice that skew lines can only
exist in three dimensions.

So, now we have considered how two
lines interact in space, letβs consider how a line and a plane might interact. Once again, we have three different
possible ways that a line and a plane can interact in space. The first way is that every point
on the line also lies on the plane. The second type of relationship
that a line and a plane can have is that they are intersecting, in which case the
intersection point is a shared, common point that lies on both of them. If the line is perpendicular to the
plane, then we can say that the line and the plane intersect orthogonally.

This brings us to the last type of
interaction, when the line and plane do not intersect. But of course because the line and
plane extend infinitely in both directions, then there is only one way in which a
line and a plane cannot be intersecting. And that is if the line and plane
are parallel.

Weβll now see the final possible
set of interactions, which will be between two planes in space. Just as we saw in the previous
geometrical interactions, there are three possible relationships between two
planes. Firstly, if the two planes share
all points, then we say that they are coincident. Secondly, we can have intersecting
planes. We could model intersecting planes
like this. As the lower diagram shows, we can
also have two planes which intersect orthogonally.

Notice that when two planes
intersect, the intersection is always a line. And of course we can have planes
that are nonintersecting. If two planes do not intersect,
they must be parallel. As a final note on intersecting
planes, here we have just been considering the intersection of two planes. But if three planes intersect, then
they can share a common point.

Weβll now see an example where we
need to identify the relationship between a given line and plane.

Observe the given figure and
choose the correct statement. Option (A) the straight line is
parallel to the plane. Option (B) the straight line is
contained within the plane. Or option (C) the straight line
intersects the plane.

In the figure, we can observe
that we have a plane, defined as π, and a line, defined with the letter πΏ. The line and the plane will
extend infinitely in all directions. We can also see that there is a
point π΄ which lies on the plane. Notice that the point π΄ also
lies on the line. Then, because the plane and the
line have a common point π΄, we can say that the straight line πΏ intersects the
plane. Therefore, the correct
statement is that given in option (C). The straight line intersects
the plane.

But before we finish this
question, it might be worth considering what the other two answer options would
look like on a diagram. Option (A) gives a statement
about a line which is parallel to a plane. A line which is parallel to a
plane can be modeled as something like this. Recall that a line and a plane
that are parallel will never intersect. But we know that in the diagram
we were given, the line and the plane intersect at point π΄. So the statement given in
answer option (A) is incorrect.

In option (B), we are
considering the statement that the line is contained within the plane. A line which is contained
within a plane might look something like this. When a line is contained within
a plane, every point on the line also lies on the plane. And it is at this point that we
can note that the way in which the original diagram was drawn is very
important. The dashed section of line πΏ
indicates that not all of the points on the line are contained on the plane
π. Therefore, the correct
statement is that the straight line intersects the plane.

We can now summarize the key points
of this video. We began by defining that a point
is a location in space. It has neither shape nor size. Then, a line is a connected set of
points that extends infinitely in two directions. A plane can be defined by three
noncollinear points or by two parallel lines or two intersecting lines. We say that a set of points are
coplanar if they lie on the same plane. If not, they are said to be
noncoplanar.

For any two coplanar lines, the
possible relationships are parallel, intersecting with an angle, or
perpendicular. For any two lines in space, the
possible relationships are parallel, intersecting with an angle, perpendicular, or
skew. For a line and plane in space, the
relationships can be intersecting with any angle, perpendicular, included in the
plane, or parallel to the plane.

Finally, we saw that two planes can
have the following configurations. They can be coincident, parallel,
intersecting at a straight line with any angle, or perpendicular. But if we are considering three
planes and not just two, they can intersect at one point, not a straight line.