Applications-of-GeoGebra/C3/3D-Geometry/English-timed

Time	Narration
00:01	Welcome to this tutorial on 3D Geometry.
00:05	In this tutorial, we will learn how to use GeoGebra to view: And construct different structures in 3D space
00:17	Solids of rotation of polynomial functions
00:21	Trigonometric functions in 3D space
00:25	Here I am using: Ubuntu Linux OS version 16.04
00:32	GeoGebra 5.0.481.0 hyphen d
00:39	To follow this tutorial, you should be familiar with:
00:43	GeoGebra interface Geometry
00:48	For relevant tutorials, please visit our website.
00:53	This image shows the rectangular coordinate system.
00:58	It is made up of mutually perpendicular axes and planes formed by them.
01:04	The axes are x (in red), y (in green) and z (in blue).
01:11	All points in 3D space are denoted by their x y z coordinates.
01:18	The point of intersection of the three axes is the origin O 0 comma 0 comma 0.
01:27	The gray rectangle in the image depicts the XY plane.
01:33	The planes divide space into 8 octants.
01:38	Point A is in the XOYZ octant and has the coordinates 4 comma 4 comma 2.
01:48	Let us draw a 3D pyramid in GeoGebra.
01:53	I have already opened a new window in GeoGebra.
01:58	This time, we work with Algebra, 2D Graphics and 3D Graphics views.
02:05	Under View, select 3D Graphics.
02:09	Click in 2D Graphics View to draw in 2D.
02:14	Drag the boundary to see 2D Graphics properly.
02:19	Click in 2D Graphics.
02:22	In 2D Graphics view, click on the Polygon tool and click on origin 0 comma 0.
02:31	This creates point A at the origin.
02:35	Then click on 2 comma 0 to create point B.
02:40	Click on 2 comma 2 for C and on 0 comma 2 to draw D.
02:48	Finally, click again on A.
02:52	Note that a quadrilateral q1 is seen in 2D and 3D Graphics views.
03:00	The length of each side is 2 units.
03:04	Click on the Move tool.
03:07	Click in 2D Graphics and drag the background.
03:11	Drag the boundary to see 3D Graphics properly.
03:16	Click in 3D Graphics and under Pyramid, on the Extrude to Pyramid or Cone tool.
03:25	In 3D Graphics view, click on the square.
03:29	In the Altitude text-box that opens, type 3 and click OK.
03:36	A pyramid e appears in 3D Graphics view.
03:40	Its base is the quadrilateral q1.
03:44	Its apex is E 1 comma 1 comma 3.
03:49	Its altitude or height is 3 units.
03:54	Rotation of a Polynomial
03:57	Let us rotate f of x equals minus 2 x raised to 4 minus x cubed plus 3 x squared.
04:07	We will rotate the part that lies in the second quadrant, in XY plane, about the x-axis.
04:16	I have already opened a new window in GeoGebra.
04:21	We will initially work with Algebra and 2D Graphics views and open 3D Graphics view later.
04:29	In the input bar, type the following line.
04:33	To type the caret symbol, hold Shift key down and press 6.
04:36	Spaces here denote multiplication. Press Enter.
04:46	Under Perpendicular Line, click on Parallel line and on the y-axis.
04:54	Keep the cursor on the x-axis.
04:58	Drag it along until you see function f, x-axis at the intersection of f and x-axis.
05:07	Click on this intersection point.
05:10	Point A appears.
05:13	Click on Slider and in Graphics view.
05:18	A Slider dialog-box opens.
05:21	Leave a as the Name.
05:24	Change Min value to minus 1.5, Max value to 0 and Increment to 0.05.
05:34	Click OK.
05:36	This creates slider a, which changes the value of a from minus 1.5 to 0.
05:45	It will focus on the part of the graph in the second quadrant.
05:51	In the input bar, type the following in parentheses.
05:55	a comma f a in parentheses. Press Enter.
06:02	This creates point B whose x coordinate is the value of a.
06:09	Its y-coordinate lies along the curve described by the function f between x equals 1.5 and 0.
06:19	Right-click on slider a and check Animation On.
06:25	Point B travels along function f as a changes.
06:31	Right-click on slider a and uncheck Animation On.
06:37	In the input bar, type a comma 0 in parentheses and press Enter.
06:47	This creates point C.
06:50	As its x co-ordinate a changes, C moves below point B along the x-axis.
06:58	Under Line, click on Segment and click on B and C to join them.
07:07	Click on Move Graphics View and drag the background to the left.
07:13	Click on View and check 3D Graphics to see the 3D Graphics view.
07:20	Note that what is drawn in 2D Graphics appears in the XY plane, in 3D Graphics.
07:27	Click in 3D Graphics view and on Rotate 3D Graphics View.
07:34	Rotate 3D Graphics to see the curve properly.
07:41	Place the cursor on the y-axis in green.
07:46	Click to see an arrow aligned with the y-axis.
07:51	Drag to pull the y-axis in or outwards to see the curve.
07:58	In the input bar, type the following line.
08:02	This creates circle c with center at point C.
08:07	Its radius is equal to f of a corresponding to the value of a on slider a.
08:15	Its rotation is around the x-axis. Press Enter.
08:21	In Algebra view, right-click on circle c and check Trace On option.
08:28	Right click on slider a and select Animation On option.
08:35	Observe the solid traced as a changes.
08:39	Watch both 2D and 3D Graphics views.
08:44	Segment BC moves between the x-axis and function f.
08:50	The part of function f that is in the second quadrant in 2D, rotates around the x-axis.
08:58	Drag 3D Graphics to see it from another angle.
09:03	Finally, let us look at trigonometric functions in 3D.
09:09	I have already opened a new window in GeoGebra.
09:14	Under View, click on 3D Graphics.
09:19	Drag the boundary to see 2D Graphics properly.
09:23	Click in 2D Graphics, then on the Slider tool and in Graphics view.
09:32	A slider dialog-box opens.
09:35	By default, the Number radio-button is selected. In the Name field, type t.
09:43	Set Min to minus 6, Max to 16 and increment of 0.1. Click OK.
09:54	This creates a slider t which will change t from minus 6 to 16.
10:01	In the input bar, type f t in parentheses equals cos t in parentheses and press Enter.
10:12	Click in 2D Graphics.
10:15	Under Move Graphics View, click on Zoom Out and click in 2D Graphics.
10:23	Click on Move Graphics View and drag the background.
10:28	You can see the graph of the cosine function of f of t, in 2D and 3D Graphics views.
10:37	Similarly, in the input bar, type g t in parentheses equals sin t in parentheses. Press Enter.
10:49	Sine function graph of g of t appears.
10:53	In the input bar, type h t in parentheses equals t divided by 4 and press Enter.
11:05	Line h of t is of the form y equals mx where slope m is 1 divided by 4.
11:14	Click in 3D Graphics view.
11:17	Click on the Point tool and click in the gray area in 3D Graphics view. This creates point A.
11:26	Drag the boundary to see its co-ordinates properly.
11:30	In Algebra view, double-click on A.
11:34	Change the coordinates to the following. Press Enter.
11:39	The x- coordinate of A is cos t.
11:44	The y-coordinate is sin t and t divided by 4 is its z coordinate.
11:53	Right-click on slider t and click on Object Properties.
11:58	A Preferences dialog-box opens.
12:02	Click on Slider tab.
12:05	Under Animation, for Repeat, choose option “Increasing” from the dropdown menu.
12:12	Close the Preferences dialog box.
12:15	In Algebra view, right-click on A and select Trace On.
12:22	Right-click on slider t and check Animation On.
12:27	Point A traces a helix in 3D space with coordinates mentioned earlier.
12:34	Click in Rotate 3D Graphic View and rotate the background.
12:39	Rotate 3D Graphics view so you are looking down the z-axis at the XY plane.
12:46	Note that the traces of A are the circumference of a unit circle.
12:52	Point A moves along the circle as angle t changes.
12:58	In 2D, its coordinates are cos t comma sin t.
13:05	Let us summarize.
13:07	In this tutorial, we have learnt how to use GeoGebra to view:
13:13	And construct different structures in 3D space
13:17	Solids of rotation of polynomial functions
13:21	Trigonometric functions in 3D space
13:25	As an assignment: Construct a prism and a cylinder anywhere in 3D space.
13:33	Draw lines to pierce the structures and find their intersection points.
13:39	Graph the given polynomial.
13:42	Show the solid formed due to rotation of the peak, in the first quadrant, in the XY plane.
13:50	As another assignment, You tried to fly a kite off a cliff. The kite got dumped into the lake below.
13:59	You gave out 325 feet of string.
14:03	The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees. How high is the cliff?
14:13	The video at the following link summarizes the Spoken Tutorial project. Please download and watch it.
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14:51	This is Vidhya Iyer from IIT Bombay, signing off. Thank you for joining.

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