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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


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.

14:21 The Spoken Tutorial project team: conducts workshops using spoken tutorials and

gives certificates on passing online tests.

14:31 For more details, please write to us.
14:34 Please post your timed queries on this forum.
14:38 Spoken Tutorial project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

14:51 This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

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