Difference between revisions of "Applications-of-GeoGebra/C3/3D-Geometry/English-timed"

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{|border=1
 +
||'''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.
 +
|-
 +
||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.
 +
|-
 +
|}

Latest revision as of 12:38, 21 October 2020

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.

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.

Contributors and Content Editors

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