WEBVTT
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In this video, we will learn to identify and model geometric concepts, points, lines, and planes in space.
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Weβll also consider their properties and what happens when they interact with one another.
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Before we start, letβs remind ourselves what we mean when we say points, lines, and planes.
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In geometry, a point is a location.
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It has neither shape nor size.
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However, when weβre working with the concept of a point, we have to represent it somehow.
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And so, we represent it with a dot, and itβs generally named by a capital letter.
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If you saw something like this, you would call it the point π΄.
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In algebra, we consider the location of points in two dimensions, along the π₯- and π¦-axis, which would give point π΄ an π₯-coordinate and a π¦-coordinate.
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But when weβre talking about a point in space, it has a third dimension of π§, which means that the coordinates of π΄ in space will be made up of three components: an π₯, a π¦, and a π§.
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For the purposes of this video, we wonβt be labeling our points, but itβs good to remember that those do exist.
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A line is a straight set of points that extend infinitely in two directions.
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If we have two points π΄ and π΅, there is exactly one line that passes through both points.
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Any other routes to get from π΄ to π΅ would have to include a curve.
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We use arrows on either end of the line to indicate that it extends infinitely in two directions.
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One way to name a line is by giving them a name of any two points that fall on that line.
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In this case, we could call this line π΄π΅.
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Sometimes, lines are named with a variable.
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When this happens, theyβre often lowercase script letters.
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And then, this line could be called line π.
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The written notation for the line π΄π΅ would be the capital letters π΄π΅ with a small line above it.
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It would also be correct to write line π.
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If weβre specifically talking about the space from point π΄ to point π΅, it does not fit the definition of a line, and that is why we would call it a line segment.
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A line segment has two end points, and a line continues in both directions.
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If you see π΄π΅ with a line above it that does not have arrows, that is indicating a line segment.
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And it tells us that it does not extend infinitely in either direction.
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We should also note that any points that fall on the same line are called colinear points.
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And if you have a point that does not fall on the same line, then the two points are said to be noncolinear points.
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The final object in space we want to consider is a plane.
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A plane is a flat surface made up of points that extend infinitely in all directions.
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When weβre sketching a plane, we usually use a quadrilateral to represent that.
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The key property we need to remember about a plane is that there is exactly one plane through any three noncolinear points.
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We also know that points or lines that fall on the same plane can be called coplanar, while points or lines that do not fall on the same plane would be called noncoplanar.
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We can label a plane with any three of the noncolinear points on that plane, here, plane π΄π΅πΆ.
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But we should also note that the order of the capital letters does not matter.
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We could call this plane plane π΅π΄πΆ.
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You might also see a plane labeled with a capital script letter.
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In this case, we could call it plane πΎ.
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Now, letβs consider the way that these points lines and planes can interact with each other in space.
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When it comes to lines in space, we can categorize their relationships in two different ways, lines that intersect and lines that do not intersect.
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For any two intersecting lines in space, they can be found on the same plane.
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We can say this because for any pair of intersecting lines, we know that weβll have at least three noncolinear points Here, π΄ is not found on the line through πΆπ·.
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By showing that at least three noncolinear points are here, we can say they exist in the same plane.
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What about lines that do not intersect?
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Almost immediately, you probably thought of parallel lines because we know that parallel lines do not intersect.
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Another key property of parallel lines is that they are the same distance from one another at every point.
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If we think about some points on parallel lines, we recognize that in a pair of parallel lines, there exist at least three noncolinear points.
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And therefore, for a set of parallel lines in space, they will fall on the same plane.
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Weβve just shown that in space parallel lines and intersecting lines are coplanar.
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Weβre used to having just two categories of lines, intersecting or parallel.
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However, in space, we have a third category.
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Lines that do not intersect and are noncoplanar are called skew lines.
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And hereβs what a pair of skew lines might look like.
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These two lines would not intersect because the line β occurs in a vertically higher plane than the line π.
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However, they do not fit the definition of parallel lines because at some points line π is closer to line β than at other points, making skew lines, lines that do not intersect and do not fall on the same plane.
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Letβs take the properties weβve just looked at and answer some questions about the intersections of points, lines, and planes.
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How many planes can pass through three noncolinear points?
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Imagine that we have three arbitrary points in space π΄, π΅, and πΆ.
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We know that between any two points, there exists only one line, which means one line passes through the point π΄π΅; one line could pass through the point π΄πΆ, which would create a set of intersecting lines.
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And if there wasnβt a line passing from the point π΄πΆ, if the line through πΆ is parallel to the line π΄π΅, it is still true that π΄, π΅, and πΆ are noncolinear.
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Theyβre not on the same line.
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But parallel lines in space and intersecting lines in space are coplanar; they exist on the same plane.
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And this means that through any three non colinear points, there will be exactly one plane.
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Based on the properties of points, lines, and planes in space, we can say that there exists exactly one plane through any three noncolinear points.
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In our next example, we need to find the intersection between a line segment and a plane.
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What is the intersection between segment π΅π΅ prime and the plane π΄π΅πΆ?
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Letβs start by identifying the plane π΄π΅πΆ.
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This will be the plane that contains the three noncolinear points, π΄, π΅, and πΆ.
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It, of course, would include this triangular piece, but it also extends in all directions.
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Next, we can identify the segment π΅π΅ prime, which is here.
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The line π΅π΅ prime would extend in both directions, but weβre only interested in the segment between π΅ and π΅ prime.
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When we look for the intersection, weβre looking for points that are common to both objects.
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And in this case, thatβs only π΅.
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The plane π΄π΅πΆ and the segment π΅π΅ prime only share one point, point π΅.
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You could maybe imagine this a bit like if you balanced a pencil on top of a flat sheet of paper, where the pencil represents a line segment and the piece of paper represents a plane.
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The intersection there would be a single point.
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And that single point in the case of this object would be point π΅.
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In our next example, weβll consider the intersection of two planes.
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What is the intersection of the plane through π΄π΅π΅ prime π΄ prime and the plane through π΅πΆπΆ prime π΅ prime?
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Weβve been given four points to identify our plane.
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We see that π΄π΅π΅ prime π΄ prime represents a face on this rectangular prism.
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However, the plane that passes through this point would be larger than this, as this plane extends in all directions.
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Next, we need to identify the plane that would pass through π΅πΆπΆ prime π΅ prime.
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Again, we have a plane that is passing through a face on a rectangular prism.
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But again, we know that these planes extend infinitely in all directions.
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By extending that π΅πΆπΆ prime π΅ prime plane, we see that it is slicing through the π΄π΅π΅ prime π΄ prime plane, and the intersection is happening at the line that passes through the points π΅π΅ prime.
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The intersection of these two planes form a line, and we can name this line π΅π΅ prime.
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In our next example, weβll see what can happen at the intersection of three different planes.
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What is the intersection of the planes ππ΄π΅, ππ΅πΆ, and ππ΄πΆ?
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First, letβs look at ππ΄π΅.
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The plane ππ΄π΅ would be the plane that contains this face in our triangular pyramid.
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If we do the same thing for the points ππ΅πΆ, itβs here.
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If we were just speaking of the intersection of the plane ππ΄π΅ and ππ΅πΆ, that intersection is the line containing the points π and π΅.
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If we add in this final plane ππ΄πΆ, this space here, we then end up with an intersection of the plane ππ΄π΅ and ππ΄πΆ at the line ππ΄.
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And we see the plane ππ΅πΆ intersects the plane ππ΄πΆ at the line ππΆ.
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But now, we need to consider what is the shared space from all three of these planes.
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And the only shared space among all three of these planes is the point π.
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If we visualized this pyramid from the top down, again, we can see that the common space to all three of these planes is the point π.
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Letβs consider one final example of the intersection of two planes.
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π and π are planes that intersect at line πΏ.
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π΅ is a point on plane π, and πΆ is a point on plane π.
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Determine the intersection of plane π with plane π΄π΅πΆ.
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In our figure, we can see line πΏ is the intersections of plane π and π.
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We want the intersection of plane π with plane π΄π΅πΆ.
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Letβs first identify plane π.
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We can see that the plane π includes the points π΄ and πΆ.
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And if weβre interested in plane π΄π΅πΆ, we recognize that the points π΄ and π΅ are located along the same plane, plane π, and the points πΆ and π΄ are both located in the plane π.
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This tells us that the point π΄ is located in plane π and in plane π.
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Because the points π΄, π΅, and πΆ are noncolinear, they can form exactly one plane.
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To see plane π΄π΅πΆ, first, weβll connect a line through points π΄ and π΅ and another line through the points π΄, πΆ.
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From there, we can add an additional line that goes through point πΆ.
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By connecting these lines, we have some idea of how this plane would look.
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At this point, we should start to see that the plane π΄π΅πΆ is slicing through the plane π.
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And this slicing is happening along the line that goes through the points π΄ and πΆ.
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Plane π contains the line π΄πΆ, as does plane π΄π΅πΆ, which makes the intersection of plane π and plane π΄π΅πΆ the line π΄πΆ.
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Before we finish, letβs quickly review our key takeaways from this video.
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For any two lines in space, the possible configurations will be parallel, intersecting, or skew.
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A plane can be defined by three noncolinear points or two intersecting lines.
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And finally, for a line and a plane in space, the possible configurations will be intersecting at a point, line included in the plane, or line parallel to the plane but not included.