THE BEGINNING OF 3-SPACE
So now we start dealing with not just (x,y) but (x, y, z).Note that even when we draw 3-D graphs in 2-D on a paper and it looks like one of the axes are diagonal on the page, it's actually in a right angle in all dimensions.
2-space: quadrants
3-space: OCTANTS
The first octant is where (x, y, z) are all positive.
We have a right hand rule for graphing in 3-space:
Point hand (tips of fingertips) in x-direction
Curl fingers towards y-axis
Thumb subsequently shows z-direction
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| picture of what our cube looked like on the white board |
We did an example where we graphed on our paper a cube of side length 4 with a geometric center at the origin, and were asked to label the vertices. See figure 13.1.3 in the book.
Then we adapted the Pythagorean Theorem into the Distance Formula.
But you can also use the Distance Formula in 3-Space...
And then we took a trip down memory lane to recall Completing the Square.
We were asked to find the center and radius of the circle given by the equation
note: always write equations by order of power, starting with
x2 and followed by y2, etc. This is called standard form
Now we're going to apply Completing the Square to SPHERES.
The parent form of the Sphere equation looks like this:
where
x0 , y0 , and z0 are the point values of the center of the sphere and r is the radius.
Let's do an example:
Find the center and radius of the sphere given by the equation
Then we learned about this word called extrusion.
extrusion: a translation of the 2-D graph in the direction of the "missing variable".
some quick Grapher sketches showing extrusion:
please note that we couldn't figure out how to show the axes labels in Grapher, so you will have to figure out which axis is which on your own! Sorry.
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| y=x |
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| y=z2 |
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| z=1 |
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z=siny |
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| x=z |
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| x2+z2=1 |
Okay, that's all! Enjoy!
-Gabe & KT
