View source for Applications-of-GeoGebra/C3/3D-Geometry/English-timed

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||'''Time'''
||'''Narration'''

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||00:01
||Welcome to this tutorial on '''3D Geometry'''.

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||In this '''tutorial''', we will learn how to use '''GeoGebra''' to view:

And construct different structures in 3D space

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||Solids of rotation of polynomial functions

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||Trigonometric functions in 3D space
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||Here I am using:

'''Ubuntu Linux''' OS version 16.04

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||'''GeoGebra''' 5.0.481.0 hyphen d
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||To follow this '''tutorial''', you should be familiar with:

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||'''GeoGebra''' interface

Geometry

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||For relevant '''tutorials''', please visit our website.

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||This image shows the '''rectangular coordinate system'''.

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||It is made up of mutually perpendicular axes and planes formed by them.

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||The axes are '''x''' (in red), '''y''' (in green) and '''z''' (in blue).

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||All points in 3D space are denoted by their x y z coordinates.

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||The point of intersection of the three axes is the '''origin O 0 comma 0 comma 0'''.

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||The gray rectangle in the image depicts the '''XY''' plane.

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||The planes divide space into 8 octants.

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||Point '''A''' is in the '''XOYZ''' octant and has the '''coordinates 4 comma 4 comma 2'''.
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||Let us draw a 3D pyramid in '''GeoGebra'''.

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||01:53
||I have already opened a new window in '''GeoGebra'''.

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||This time, we work with '''Algebra, 2D Graphics''' and '''3D Graphics''' views.
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||Under '''View''', select '''3D Graphics'''.
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||Click in '''2D Graphics View''' to draw in '''2D'''.
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||Drag the boundary to see '''2D Graphics''' properly.
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||Click in '''2D Graphics'''.
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||In '''2D Graphics''' view, click on the '''Polygon''' tool and click on origin 0 comma 0.

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||This creates point '''A''' at the origin.
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||Then click on 2 comma 0 to create point '''B'''.
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||Click on 2 comma 2 for '''C''' and on 0 comma 2 to draw '''D'''.
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||Finally, click again on '''A'''.
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||Note that a quadrilateral '''q1''' is seen in '''2D''' and '''3D Graphics''' views.

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||The length of each side is 2 units.
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||Click on the '''Move''' tool.
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||Click in '''2D Graphics''' and drag the background.
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||Drag the boundary to see '''3D Graphics''' properly.
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|| Click in '''3D Graphics''' and under '''Pyramid''', on the '''Extrude to Pyramid or Cone''' tool.

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||In '''3D Graphics''' view, click on the square.
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||In the '''Altitude''' text-box that opens, type 3 and click '''OK'''.
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||A pyramid '''e''' appears in '''3D Graphics''' view.

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||Its base is the quadrilateral '''q1'''.

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||Its apex is '''E''' 1 comma 1 comma 3.

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||Its altitude or height is 3 units.
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||'''Rotation of a Polynomial'''

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||Let us rotate '''f of x''' equals minus '''2 x raised to 4''' minus '''x cubed''' plus '''3 x squared'''.

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||We will rotate the part that lies in the second quadrant, in '''XY''' plane, about the '''x-axis'''.
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||I have already opened a new window in '''GeoGebra'''.

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||We will initially work with '''Algebra''' and '''2D Graphics''' views and open '''3D Graphics''' view later.

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||In the '''input bar''', type the following line.

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||To type the '''caret symbol''', hold '''Shift''' key down and press 6.

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||Spaces here denote multiplication.

Press '''Enter'''.
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||Under '''Perpendicular Line''', click on '''Parallel line''' and on the '''y-axis'''.
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||Keep the '''cursor''' on the '''x-axis'''.
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||Drag it along until you see '''function f, x-axis''' at the intersection of '''f''' and '''x-axis'''.
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||Click on this intersection point.
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||Point '''A''' appears.
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||Click on '''Slider''' and in '''Graphics''' view.
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||A '''Slider''' dialog-box opens.
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||Leave '''a''' as the '''Name'''.
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|| Change '''Min''' value to '''minus''' 1.5, '''Max''' value to 0 and '''Increment''' to 0.05.
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||Click '''OK'''.
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||This creates '''slider a''', which changes the value of '''a''' from minus 1.5 to 0.

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||It will focus on the part of the graph in the second '''quadrant'''.

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||In the '''input bar''', type the following in parentheses.

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||'''a''' comma '''f a''' in parentheses.

Press '''Enter'''.
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||This creates point '''B''' whose '''x coordinate''' is the value of '''a'''.

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||Its '''y-coordinate''' lies along the curve described by the '''function f''' between '''x''' equals  1.5 and 0.
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||Right-click on '''slider a''' and check '''Animation On'''.
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||Point '''B''' travels along '''function f''' as '''a''' changes.
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||Right-click on '''slider a''' and uncheck '''Animation On'''.
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||In the '''input bar''', type '''a''' comma 0 in parentheses and press '''Enter'''.
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||This creates point '''C'''.
 
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||As its '''x co-ordinate a''' changes, '''C''' moves below point '''B''' along the '''x-axis'''.
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||Under '''Line''', click on '''Segment''' and click on '''B''' and '''C''' to join them.
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||Click on '''Move Graphics View''' and drag the background to the left.
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||Click on '''View''' and check '''3D Graphics''' to see the '''3D Graphics''' view.
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||Note that what is drawn in '''2D Graphics''' appears in the '''XY''' plane, in '''3D Graphics'''.
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||Click in '''3D Graphics''' view and on '''Rotate 3D Graphics View'''.

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||Rotate '''3D Graphics''' to see the curve properly.

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||Place the '''cursor''' on the '''y-axis''' in green.

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||Click to see an arrow aligned with the '''y-axis'''.

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||Drag to pull the '''y-axis''' in or outwards to see the curve.

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||In the '''input bar''', type the following line.

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||This creates circle '''c''' with center at point '''C'''.

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||Its radius is equal to '''f of a''' corresponding to the value of '''a''' on '''slider a'''.

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||Its rotation is around the '''x-axis'''.

Press '''Enter'''.
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||In '''Algebra''' view, right-click on circle '''c''' and check '''Trace On''' option.
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||Right click on '''slider a''' and select '''Animation On''' option.
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||Observe the solid traced as '''a''' changes.
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||Watch both '''2D''' and '''3D Graphics''' views.
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||Segment '''BC''' moves between the '''x-axis''' and '''function f'''.
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||The part of '''function f''' that is in the second '''quadrant''' in 2D, rotates around the '''x-axis'''.
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||Drag '''3D Graphics''' to see it from another angle.
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||Finally, let us look at '''trigonometric functions''' in 3D.

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||I have already opened a new window in '''GeoGebra'''.
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||Under '''View''', click on '''3D Graphics'''.
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||Drag the boundary to see '''2D Graphics''' properly.
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||Click in '''2D Graphics''', then on the '''Slider''' tool and in '''Graphics''' view.
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||A '''slider''' dialog-box opens.

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||By default, the '''Number''' radio-button is selected.

In the '''Name''' field, type '''t'''.
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||Set '''Min''' to minus 6, '''Max''' to 16 and '''increment''' of 0.1.

Click '''OK'''.
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||This creates a '''slider t''' which will change '''t''' from minus 6 to 16.  
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||In the '''input bar''', type '''f t''' in parentheses equals '''cos t''' in parentheses and press '''Enter'''.
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||Click in '''2D Graphics'''.
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||Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''2D Graphics'''.
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||Click on '''Move Graphics View''' and drag the background.
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||You can see the graph of the '''cosine function''' of '''f of t''', in '''2D''' and '''3D Graphics''' views.
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||Similarly, in the '''input bar''', type '''g t''' in parentheses equals '''sin t''' in parentheses.

Press '''Enter'''.
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||'''Sine function''' graph of '''g of t''' appears.   
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||In the '''input bar''', type '''h t''' in parentheses equals '''t''' divided by 4 and press '''Enter'''.
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||Line '''h of t''' is of the form '''y''' equals '''mx''' where slope '''m''' is 1 divided by 4.
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||Click in '''3D Graphics''' view.
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||Click on the '''Point''' tool and click in the gray area in '''3D Graphics''' view.

This creates point '''A'''.
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||Drag the boundary to see its '''co-ordinates''' properly.
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||In '''Algebra''' view, double-click on '''A'''.

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||Change the '''coordinates''' to the following. Press '''Enter'''.
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||The '''x- coordinate''' of '''A''' is '''cos t'''.

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||The '''y-coordinate''' is  '''sin t''' and '''t''' divided by '''4''' is its '''z coordinate'''.
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||Right-click on '''slider t''' and click on '''Object Properties'''.
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|| A '''Preferences''' dialog-box opens.

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||Click on '''Slider''' tab.

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||Under '''Animation''', for '''Repeat''', choose option “'''Increasing'''” from the dropdown menu.
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||Close the '''Preferences''' dialog box.
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||In '''Algebra''' view, right-click on '''A''' and select '''Trace On'''.
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||Right-click on '''slider t''' and check '''Animation On'''.
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||Point '''A''' traces a '''helix''' in '''3D''' space with '''coordinates''' mentioned earlier.
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||Click in '''Rotate 3D Graphic View''' and rotate the background.

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||Rotate '''3D Graphics''' view so you are looking down the '''z-axis''' at the '''XY''' plane.
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||Note that the traces of '''A''' are the circumference of a '''unit circle'''.

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||Point '''A''' moves along the circle as angle '''t''' changes.

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||In '''2D''', its '''coordinates''' are '''cos t''' comma '''sin t'''.
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||13:05
||Let us summarize.
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||13:07
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to view:

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||13:13
||And construct different structures in 3D space

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||13:17
||Solids of rotation of polynomial functions

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||13:21
||Trigonometric functions in 3D space
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||13:25
||As an assignment:

Construct a prism and a cylinder anywhere in 3D space.

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||Draw lines to pierce the structures and find their intersection points.

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||Graph the given '''polynomial'''.

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||Show the solid formed due to rotation of the peak, in the first '''quadrant''', in the '''XY''' plane.
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|| As another assignment,

You tried to fly a kite off a cliff. The kite got dumped into the lake below.

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||You gave out 325 feet of string.

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||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?
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||14:13
||The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
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||14:21
||The '''Spoken Tutorial '''project''' '''team: conducts workshops using spoken tutorials and

gives certificates on passing online tests.

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

More information on this mission is available at this link.
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||14:51
||This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
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