Goal:
To understand (systems of) linear equations in terms of geometry in the
Euclidean plane, and vice versa.
Program:
1. Start matlab and run your matlab script setpath to add the guis/ascii and guis/binaries folders to your matlab search path.
2. Give the command open('RM+.fig') to open the Road Map to the Mathematics
Laboratory, and press the Euclids P button to step out in the Euclidean plane.
Alternatively you may enter this plane directly from the matlab prompt by the
command open('EUCLID.fig').
3. Use Euclid's lab as follows:
First click at the clear button. Then click at the coords button, and study
how the coordinates of the cursor point changes as you move the cursor around
in the left window.
The define a line in the plane by giving first a point on the line (in the
x- edit box), and then a vector giving the direction of the line (in the
// to edit box), followed by a return. A line should now appear with the
given directional vector in red starting from the given point on the line.
You can then edit the given line by point dragging (with the left mouse button
down), on a point near the given point on the line to translate the line,
or a point near the arrow head to rotate the line. Watch how the equation
defining the line changes as you redefine the line.
Now try defining a line by pressing the click line button and then click at
two points of the desired line. If there is no "handle" (directional vector)
to the new line, you can get one by pressing the handles (on) button. You
can also shift between a parallell vector or an orthogonal vector defining
the direction of the line using the normal/tangent button.
With two line present on the screen you should also see a black (or possibly blue) spot with
given coordinates where the two lines intersect. This point represents the
solution of the corresponding two linear equations displayed in the top right
window, which you can toggle between vector form and component form.
The vector form of this system of equation is also illustrated graphically
in the lower right window, where the two vectors multiplied by x1 and x2,
respectively, is colored in grey, and the target vector on the right is
colored black. You can not edit these vectors directly, but study how they
change as you change the two lines to the left. Make sure you understand
what is happening and the (rather intricate) interrelation between the three
different representations, the formulas/equations in the top right window,
the lines to the left, and the vectors to the lower right.
You may solve the system of equations in the top right window several ways,
by hand calculation using variable elimination, just by looking (at the
coordinates of the intersection point) in the left window, or (as a sport)
by seeking the combination of the two grey column arrows, colouered black,
that matches the target column vector to the right, coloured purple. To do
this you should first press the lines off button and then the trial solution
button. You then move the cursor around with the left mouse button down in
the left window until the corresponding black combination of the two grey
column arrows matches the purple target vector of the right hand side of the
system of equations. To check your answer you may then press the solve
equations button.
No its time for some exercises!
4. Delete all but one line and change this by translation and by rotation,
respectively. In what way does the equation for the line change as you
a) translate b) rotate the line. What is the relation between the orthogonal
arrow handle to a line and the coefficients of the corresponding equation?
5. Create two lines which intersect in the point ( 0.6, -.3).
6. In which point do the lines x - [ 0, 1] // [ 1, -1] and x - [ -1, -1] // [ 1, 0.5]
intersect?
7. What is the corresponding ax + by = c representation of these two lines?
Are these representations unique, for given lines? Again, what is the relation
between the equations in the top right window and the "handles" defining
the line directions/normals?
8. Solve the system of equations by hand calculation. Does your solution
match the coordinates of the intersection point? Why/why not?
9. Under what condition do two lines (in a plane) intersect?
10. Can two lines in a plane have more than one intersection point?
11. Now create a third line. Is there a common intersection point of the
three lines? Could it be? Is it most natural to have or not to have such
a common intersection point of more than two lines?
12. How do you interprete the black spot with coordinates in the left window
in the present case with more lines than two/more conditions/equations
than variables to satisfy the equations? You may also try solving the
corresponding set of three equations by hand, or by trying to combine the
two grey column vectors to a given target in three dimensional space! What is the reason the combination can seem to match the target, but yet miss?
/Kenneth