DO NOT DO THE FIRST PROBLEM! To gain more familiarity with calculations in R 3 ,
ID: 3011706 • Letter: D
Question
DO NOT DO THE FIRST PROBLEM!
To gain more familiarity with calculations in R3, let us pursue the example of four “points” given above.
5.3.1 Find the plane ax+by+cz = 0 through the points (0,0,1) and (1,1,1), and check that it does not contain the points (1,0,0) and (0,1,0).
5.3.2 Show that RP2 has four “lines,” no three of which have a common “point.” Not only does RP 2 contain four “lines,” no three of which have a “point” in common; the same is true of any projective plane, because this property follows from the projective plane axioms alone.
5.3.3 Suppose that A,B,C,D are four “points” in a projective plane, no three of which are in a “line.” Consider the “lines” AB,BC,CD,DA. Show that if AB and BC have a common point E, then E = B.
5.3.4 Deduce from Exercise 5.3.3 that the three lines AB,BC,CD have no common point, and that the same is true of any three of the lines AB,BC,CD,DA.
Explanation / Answer
As ax+by+cz=0 passes through the point (0,0,1),
a*0+b*0+c*1=0
=> c=0
So the equation becomes ax+by =0 and it passes through (1,1,1).
So a+b=0
=> a= -b
Thus given equation becomes
-bx+ by=0
=> -b(x-y)=0
=> x-y=0
=> x=y which is the required equation of plane.
Now (1,0,0) does not belong to the plane x=y as its x coordinate =1 and y coordinate =0 and they are not equal.
(0,1,0) does not belong to the plane x=y as its x coordinate =0 and y coordinate =1 and they are not equal.
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