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When
the graphic box is clicked on a three dimensional grid will appear.
You will not be able to see the z-axis because it is coming out
at you. Go to view 2 and the axis will be rotated so you may view
the grid in 3-Dimensions. Click on any point and notice the lower
left hand corner; a set of three coordinates appear (x,y,z).
What
do you think the z now represents?
To get a better feel for 3-D geometry click on cube at the right
side of the menu box. Click on "label cube" , the letters
A,B,C,D will appear.
**What
are the x,y,z coordinates for A______ B______ C_____ D____
**What is the measure of line segment AB? _____
When
you have completed click off the cube and label cube
n tutorial II
you used Kinemage to measure angles and distances in two dimensions.
Because the real world is 3- Dimensions, lets take a look at measuring
angles and distances in 3-D space. When measuring angles in space
we will need to click four consecutive points. Can you explain why
three point will not be enough?
(Optional:
In solid geometry the dihedral angle is defined as he angle formed
by the intersection of two planes. A dihedral angle may be regarded
as formed by the rotation of a plane about any line in the plane
of an axis. Thus the value of a dihedral angle depends upon the
amount of rotation about the edge, and not really upon the extent
of the planes.)
Click on
two planes, and "label plane". Go to measure under the
Other pulldown. Just as you measured angles in Tutorial II, click
point A, then B, then point C. This will give you the measure
of <ABC. Now click on point D you will have measured the dihedral
angle ABCD (see dhr= on the lower right of the black Mage box.
Much like
angles shown in tutorial 2 required three points located on a
two dimensional plane, the dihedral angles demonstrated here requires
four points in three dimensional space.
**What is
the measure of <ABC?_____
**What is the measure of the dihedral angle ABCD formed by the
intersection of the two planes?______
Clear the
screen of the planes and labels. Return to View 2. Choose dihedral
1, click on label angle. Click consecutively points A,B,C.D notice
the lower right gives the angle between the last three points
and that dhr= gives the dihedral angle for ABCD.
Go back to
View 1, it shows the dihedral angle face on. Measure the angle
as you see it, face on. What is this telling you? Rotate the image
about several axis to gain a better understanding of a dihedral
angle.
Clear the
screen of dihedral angle 1. Repeat using the other sample dihedral
angles (no letters will appear). Try to rotate the axis until
you gain a clear understanding of dihedral angles. You may also
use this module to form three dimensional images. These images
may be saved and used on your own databases. (To save images will
require the use of text editor, so check with your teacher first).
Activities
for Students:
1) In tutorial
II we used the Pythagorean theorem to find the diagonal of a square
(a^2+b^2=c^2), and then checked the results using Kinemage. What
equation would represent the diagonal of a cube? Why? Test out your
theory and them use Kinemage to check you reasoning.
2)Try creating
several 3-Dimensional figures using the 3-D grid. It may be helpful
to be in either view2 or view3 when this is done.
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