|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:|
|00:48||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.
|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.|
|05:36||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.
|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.
|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.
|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.
|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?
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