WEBVTT
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Determine the coordinates of point π΄.
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Hopefully, we know how to find the coordinates of a point in two dimensions, so on the plane.
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Our point π΄ however, like us, lives inside three-dimensional space.
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How do you find its coordinates?
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We can use what we know about coordinates in the plane to help us.
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The point π΅ lies in the π₯π¦-plane.
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Letβs ignore the π§-axis for a moment and forget that weβre in three-dimensional space and just focus on this π₯π¦-plane.
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We can read off the π₯-coordinate three from the π₯-axis and the π¦-coordinate negative three from the π¦-axis.
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So ignoring the third dimension, π΅ has coordinates three, negative three.
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You can think of these coordinates as instructions tell you how to get to π΅ from the origin.
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Starting at the origin, the π₯- coordinate tells us how far we have to move in the positive π₯-direction.
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So parallel to the π₯-axis, we have to move three units.
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And the π¦-coordinate tells us how far we have to move in the positive π¦-direction.
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So parallel to the π¦-axis, we have to move in negative three units in the positive π¦-direction.
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So that means moving three units in the other direction.
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And we see that if we do this we do indeed get to π΅.
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This works fine in the π₯π¦-plane where we just have two dimensions and two axes.
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We can get to π΅ just fine.
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But how do we get to π΄?
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We canβt do this by just moving parallel to the π₯- and π¦-axes.
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We have to move in the π§-direction as well.
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How many units do we have to move in the π§-direction?
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We can read off the value from the π§-axis just as we did from the π₯- and π¦-axes.
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We have to move three units in the π§-direction.
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If we do this from π΅, we succeed in getting to π΄.
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So putting this all together, to get to π΄, we have to move three units in the π₯-direction.
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That gives us our π₯-coordinate, three.
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Then we have to move negative three units in the π¦- direction.
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That gives us our π¦-coordinate, negative three.
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And finally we have to move three units in this new π§-direction, giving us a π§-coordinate of three.
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We can write our answer like this: π΄ has coordinates three, negative three, three.
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As weβre working in three dimensions, there are three coordinates: the π₯-coordinate, π¦-coordinate, and the new π§-coordinate.
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A good way to find the coordinates of a point in 3D space is to look for the point directly below it in the π₯π¦-plane.
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In our case, this was a point π΅.
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The π₯- and π¦-coordinates of π΄ in 3D space were just the π₯- and π¦-coordinates of π΅ in the 2D plane.
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The third π§-coordinate told us how far π΄ was above π΅.
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Of course this wouldβve been negative if π΄ were actually below π΅.