Applications-of-GeoGebra/C3/3D-Geometry/English-timed
From Script | Spoken-Tutorial
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. |