1.14. The familiar model for Euclidean geometry is the \"Cartesian model.\" In t

Question

1.14. The familiar model for Euclidean geometry is the "Cartesian model." In that model, points are interpreted as coordinate pairs of real numbers (x, y). Lines are loosely interpreted as equations of the form Ax + By = C, but technically, there is a little bit more to it than that. First, A and B cannot simultaneously be zero. the equations Ax + By = C and A's + B,y = C, both represent the same line (in truth then, a line is represented by an equivalence class of equations). In this model, a point (x,y) is on a line Ax + By C if its coordinates make the equation true. With this interpretation, verify the axioms of incidence. Second, if A, = k A, B, = k B, and C, = kC for some nonzero constant k, then i. hakuinan tun ather points (xi, vi) and (x3, y3) if

Explanation / Answer

First of all,Euclidean geometry is a mathametics system attributed to the alexandrian Greek mathematician Euclid,which he described the elements of the geometry.Euclids method consists in assuming a small set of axioms,and deducing many other propositions from these.

HereHere,In this problem,Given the cartesian model in the Euclidean geometry.And the points in this model are interpreted as coordinate pairs real numbers(x,y).Lines are loosly interpreted as equations of the form

Ax+By=C.-->(1)

First,A and B can not simultaneously be zero and second if #A'=kA,B'=kB and C'=kC for some nonzero constant k,

Then From the above relations A=A'/k,B=B'/k and C=C'/k,Substitute these relations in the above equation (1)

(A'/k)x+(B'/k)y=C'/k.

1/k(A'x+B'y)=C'(1/k).

After proper cancellation

A'x+B'y=C'.-->(2)

So,Above two equations (1) and (2) represent the same line with some constant k.Any way,this constant is get cancelled on both sides.So,these equations are same and coordinates are (x,y) real numbers.

with the above interpretation,For every two points A,B there exists a line that contains each of the points A,B and Here exists two points (x,y) for this line.Here,A,B,C three points do not lie on the same line there exists a plane that contains each of these points.If these two points A and B of a line lie in a plane,Then every point lie equationsme plane.So,From this proof these are the axioms of incidence.

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.