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3D Geometry in Geogebra - a single vector by keralaguest


									3D in Geogebra – vector components
Paul Robinson, IT Tallaght

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Geogebra can do a pretty good job of representing 3D objects, allowing rotations and
dilations to view things from different angles and to zoom in. These notes are based on a
construction by Michele Passante (
                                px 
                                
First a bit of theory: Let P   py  be a point in 3D space. The 3D rotation matrices about
                               p 
                                z
the x, y and z axis are
      1     0        0               cos(b) 0  sin(b)         cos(c)  sin(c) 0 
                                                                                 
 Rx   0 cos(a)  sin(a)  Ry   0           1     0     Rz   sin(c) cos(c) 0 
       0 sin(a) cos(a)               sin(b) 0 cos(b)           0          0    1
                                                                                 
where a, b and c are angles between 0o and 360o. The rotation RX will rotate P in the
horizontal plane anticlockwise about the vertical z axis through an angle of a, and
similarly for Ry and Rz. If we rotate P and the x, y, z coordinate frame using R we can
interpret the result as a rotated view of the original point P. This is what we will do in
the construction which follows.
We will write a general rotation as R  Rz Ry Rx , which will allow us to rotate about the 3
axes. Note that the 3 rotations do not commute with each other, meaning that if we write
them in a different order the result will generally be slightly different! This will not
matter in terms of using rotations to view 3D objects.
                                                                          qx 
                                                                          
 After rotation the point P will have 3D coordinate RP   qy 
                                                                         q 
                                                                          z
where the q coordinates now depend on the angles a, b and c. To see what this looks like
in 2D (on the screen!) we simply want two of the coordinates of RP. If we imagine the x
axis pointing out of the screen towards us then the screen coordinates are y, z. This
                                                qy 
means we need to plot the point Q    .      q 
                                                z
This construction is OK, but it is not very flexible. As well as Q we will want to extract
some other information from the 3D point RP, and it is not easy to do in GeoGebra with
RP in this form. Instead we will start with
                    px         1       0         0
                                                  
               P   py   p x  0   py  1   p z  0   p x E x  py E y  p z E z
                   p          0        0          1
                    z                             

Now RP  px RE x  py RE y  pz RE z . We can now think of p x as the component of the
rotated P in the direction of the rotated axis RE x . If we let Wx be the 2D vector with
components the y and z coordinates of REx , and similarly for Wy and Wz then we have

              Q  pxWx  pyWy  pzWz

The vectors pxWx , pyWy and pzWz are the components of the rotated P along the rotated
axis as viewed on the screen.

We will construct the rotation of a 3D point P with its axis frame. The point P does not
change its position, just our rotated view of it changes.
Creating a 3D axis frame
1. Put on sliders for angles a, b, c and another named d which will be used to lengthen
and shorten our axes:

        Click on the slider tool then on the drawing pad. Call it a and select the angle
option. Go with the default of 0o to 360o.
Repeat for sliders b and c.
Create another slider called d with a Number value from 0.5 to 5 in steps of 0.1.
Right click the sliders (or their values in the left hand window) if you want to change their

       click on the object selection tool if you want to move the sliders around.

2. Put in unit vectors along the x, y and z axes.
In the input line at the bottom of the screen type

      E_x = {{1}, {0}, {0}}                                          press return

                                         1
                                         
This represents the column vector E x   0  . A row of numbers would be written as
                                         
{1, 0, 0}. We need a column as we want to multiply by a matrix.
Repeat the input for
       E_y = {{0}, {1}, {0}}    and               E_z = {{0}, {0}, {1}}

3. Put in the rotation matrices Rx, Ry and Rz and the dilation matrix D
In the input line type

      R_x = {{1, 0, 0}, {0, cos(a), -sin(a)}, {0, sin(a), cos(a)}}   press return
      R_y = {{cos(b), 0, -sin(b)}, {0, 1, 0}, {sin(b), 0, cos(b)}}   press return
      R_z = {{cos(c), -sin(c), 0}, {sin(c), cos(c), 0}, {0, 0, 1}}   press return

For the general rotation type

      R = R_z*R_y*R_x                                                press return

4. Create our axes
In the input line type
       V_x = R*E_x                                                   press return

This will rotate (and dilate) the unit vector E_x which is pointing along the x-axis.
Repeat for
              V_y = R*E_y          and          V_z = R*E_z
Now we need to see what that looks like on the screen. The vectors V_x, V_y and V_z are
column vectors. As mentioned in the introduction we want the 2nd and 3rd components of
our 3D vectors to create a point on the screen.
This is probably a good point to turn off labeling. Go to Options, labeling and click on No
New Objects.
In the input line type

       W_x = (Element[Element[V_x,2],1], Element[Element[V_x,3],1])

and press return.
Element[V_x,2] is the second number in V_x, which is itself a list { } consisting of 1
number. We want the “first” number in that list. The round brackets in W_x mean that
we now have a point in 2D which you should see on the screen.
Repeat for
      W_y = (Element[Element[V_y,2],1], Element[Element[V_y,3],1])
      W_z = (Element[Element[V_z,2],1], Element[Element[V_z,3],1])

In the input line type
       u = vector[d*W_x]           press return

Repeat for v = vector[d*W_y] and w = vector[d*W_z].
Hide the points W_x, W_y and W_z by clicking on the circles next to their definition in the
left hand window.
Click on the object Selection Tool then move sliders a and b to 30o, followed by moving c.
See that d makes things bigger and smaller.

5. Make the axes look a bit nicer.
Go to View and click on Axes to remove the default GeoGebra axes.
Right click on the u vector (do it in the definition in the left hand window) and go to
Properties at the bottom of the list. Use ctrl or shift to select u, v and w in the vector list
simultaneously. Set the Colour to dark blue and the Style line thickness to 5.
In the input line type –u and press return. Do the same for –v and –w.
Select these 3 new vectors as before, leave the colour on black and the line thickness as
thin but change the line style to fine dots.
We also need to label our axes. Click on the small arrow at the bottom of the Slider Tool

       and select the Insert Text option. Click on the screen anywhere and type X for the
text. Now right click the X text, go to Properties and Position. In the Starting Point box
type 2*W_x. You may use the mouse to move the X text slightly but as you move the slider
controls a, b, c and d it should follow the arrow head of the x-axis.
Repeat this text insert for Y (Starting position 2*W_y) and Z (Starting position 2*W_z).

As a final flourish type
       Polygon[d*(W_x+W_y), d*(-W_x+W_y), d*(-W_x-W_y), d*(W_x-W_y)]
And press return.
Right click each side of this polygon and click off Show Object.

      Use the Panning Tool (click it then drag on the screen) if you want to centre your
construction a little. Click the little arrow on the panning tool to bring up the Zoom In
and Zoom Out Tools (click then click the screen) if you want your picture bigger or

We can now put other objects on our axes frame like points, lines and planes and see
what they look like in 3D. We can also make geometric objects e.g. a cube made up of
corners (points) and faces (polygons).

Illustrating a Vector in 3D
In the input line type the column vector

      P = {{1}, {1}, {1}}

As discussed in the introduction this will have the screen coordinate
Q  pxWx  pyWy  pzWz . In the input line type
       p_x = Element[Element[P, 1], 1]
       p_y = Element[Element[P, 2], 1]
       p_z = Element[Element[P, 3], 1]
       Q = p_x*W_x + p_y*W_y + p_z*W_z

And press return after each line.
Hide the point Q (click the circle next to it in the left hand window).
In the input line type


We can also put in some components of this vector. The projection (shadow) of P on the
x-y plane is the first 2 terms of Q i.e. p xWx  py Wy .
In the input line type
       Q_{xy} = p_x*W_x + p_y*W_y

And hide the point Q_{xy}. Now type

      Segment[(0, 0), Q_{xy}]
      Segment[Q_{xy}, Q]

Make these two segments have the fine dotted line style and, possibly, change their
More elements on the Screen
1. Components of P along the x, y and z axes would be pxWx , pyWy and p zW respectively,
which you could also put on the picture. You could then make segments between these
points and Q.
If you want to put the component values of P on the axes then, for the x component,
       (a) Make a textbox and type the text p_x
       (b) Make the starting position p_x*W_x

Repeat for the y and z components. GeoGebra will interpret the text p_x as its numerical
You should now be able to go back to the definition of P and change the numbers to see
other vectors.
2. Create a second point P1 (with 2D point Q1) constructed like P. The line through these
points is then created using the command Line[Q, Q1].
3. Create 3 points Q1, Q2 and Q3 and join them with the polygon command. This will be
the plane through the corresponding 3D points P1, P2 and P3.

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