WEBVTT
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In the figure shown, the points π and π΄ have coordinates zero, zero, zero and seven, five, six respectively.
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Determine the coordinates of π΅ and πΆ.
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We are told that the point π is the origin with coordinates zero, zero, zero.
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Letβs put that on our diagram, like so.
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And we can do the same thing for the point π΄ which has coordinates seven, five, six like so.
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As these points are in three-dimensional space, we need three coordinates to represent their locations.
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We therefore expect that π΅ and πΆ are going to have three coordinates too.
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Before we try to work out the value of these coordinates, I think itβs helpful to go back and look at the two-dimensional case to see what these coordinates represent.
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Here is our representation of the more familiar two-dimensional space with two axes, just π₯ and π¦.
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And we have a point in this two-dimensional space, the point π with coordinates four, three.
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Now what do the values of those coordinates, four and three, tell us?
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First, itβs helpful to add in the origin π with coordinates zero, zero.
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And weβre going to see that the coordinates of the point π, four and three, tell us how to get to the point π from the origin π.
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Starting at the origin, the first coordinate, the π₯-coordinate four, tells us that we need to walk four units in the π₯-direction.
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And the second coordinate, the π¦-coordinate three, tells us that we need to move three units in the π¦-direction; that is three units parallel to the π¦-axis.
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And having done this, weβve reached our point π.
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We could just have easily done it the other way around, first moving three units in the π¦-direction and then four units in the π₯-direction.
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It doesnβt matter which direction we go first.
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Going back to our three-dimensional picture, letβs think about what the coordinates of π΄ tell us.
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Well, they tell us how to get to the point π΄ from the origin which I have marked.
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The first coordinate of the point π΄, the π₯-coordinate seven, tells us that we need to move seven units in the π₯-direction, in this case, along the π₯-axis.
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The second coordinate of the point π΄, the π¦-coordinate five, tells us that we need to move five units in the π¦-direction; that is five units parallel to the π¦-axis.
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The third coordinate is the new coordinate that we didnβt have in the two-dimensional case; it is the π§-coordinate.
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And the fact that the π§-coordinate is six tells us that we have to continue our journey by moving six units in the π§-direction; that is six units parallel to the π§-axis.
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So that is the interpretation of the coordinates of π΄.
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I should just mention at this point that weβve been using the fact that the solid in our figure is a cuboid or right-rectangular prism.
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And therefore, all these sides are parallel to one of the axes.
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For example, π·π΅ is parallel to the π¦-axis and π΅π΄ is parallel to the π§-axis.
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So now that we have clarified what the coordinates of a point in 3D space mean, weβre ready to determine the coordinates of π΅ and πΆ.
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First, letβs find the coordinates of π΅, which is the same as asking how to get to the point π΅ from the origin π.
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And we are going to need three coordinates: the first coordinate, the π₯-coordinate, tells us how far we have to go in the π₯-direction; the second, which is the π¦-coordinate, tells us how far we have to go in the π¦-direction; and the third, the π§-coordinate, tells us how far we have to go in the π§-direction.
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Well, if we look at our figure, we can see that we already have a path to π΅ from the origin; itβs through the point π·.
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From the origin π, we first travel seven units in the π₯-direction to the point π·.
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And once we are there, we just travel five units in the π¦-direction, and we get to the point π΅.
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So what are the coordinates of π΅?
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We travelled seven units in the π₯-direction, so the π₯-coordinate is seven and then five units in the π¦-direction, so the π¦-coordinate is five and then we were already at π΅, so we didnβt have to travel any distance in the π§-direction, so our π§-coordinate is zero.
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And how about the point πΆ?
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We havenβt already drawn the path to πΆ from the origin, but we notice again that we can go through π·.
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And if you recall, we said that the length of the edge π΅π΄ was six.
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And as we are in a cuboid or right-rectangular prism, the length of π·πΆ must be six also.
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So how do you get to πΆ from the origin?
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First, you travel seven units in the π₯-direction, so the π₯-coordinate is seven.
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We donβt have to move at all in the π¦-direction, and so the π¦-coordinate is zero.
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We just have to move six units in the π§-direction, so the π§-coordinate is six.
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So thatβs our answer.
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Weβve determined the coordinates of π΅ and πΆ.
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You might like to check that going in a different route from the origin π to the point π΅ or πΆ gives the same values of the coordinates as in the two-dimensional case.