y=x+1, and y=7x+1 both are distinct lines passing through (1, 2) and (3, 4) in .
In R × R, for any two distinct points A and B, there exists a unique line containing them Show th...
Duality Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct lines. Axiom 3. There is exactly 1 line with exactly 3 distinct points on it. Axiom 4. Given any 2 distinct points, there exists at least 1 line passing through the 2 points. Which of the following is the dual of Axiom 4? O a. Every line has at least 2 points on it. b. There exists at least 1 point with at least...
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are parallel if they have no point in common. Axioms: I. Given any two distinct points, there is a unique line on both. II. Each line has at least two distinct points on it. III. There exist at least three points. IV. Not all points...
using these axioms prove proof number 5 1 - . Axiom 1: There exist at least one point and at least one line Axiom 2: Given any two distinct points, there is exactly one line incident with both points Axiom 3: Not all points are on the same line. Axiom 4: Given a line and a point not on/ there exists exactly one linem containing Pouch that / is parallel tom Theorem 1: If two distinct lines are not parallet,...
With the given notes from below, answer number one please Axiom 1: there exists at least one line. Axiom 2: every line has exactly 8 points incident (passes through) to it Axiom 3: not all points are incident to the same line Axiom 4: there is no line containing all points Axiom 5: there is at least two points on one line Axiom 6: there exists at least two lines Axiom 7: there is exactly on with incident with any...
In this problem, let li be the line that passes through the points A(1,2, 4) and B(-1,3,8), and let l2 be the line with symmetric equations x +1 = 2y = 32 — 3. Parts (e) and (f) relate to the vector field F = (xy, xz, yz). (a) Show that the lines li and l2 intersect. (b) Let P be the plane that contains both lines li and lz. Find an equation for P. (c) Show that the points...
11. We will prove the following statement by mathematical induction: Let 1,2tn be n2 2 distinct lines in the plane, no two of which are parallel Then all these lines have a point in common 1. For2 the statement is true, since any 2 nonparallel lines intersect 2. Let the statement hold forno, and let us have nno 1 inesn as in the statement. By the inductive hypothesis, all these lines but the last one (i.e. the nes 1,2.n-1) have...
(1) (a) Find the equation of the line, Li, which passes through the points A : (4,y,z) = (0, -5, -3) and B : (x, y, z)=(3, 1,0). (b) Find the equation of the line, Ly, which passes through the points C:(x, y, z)=(-1, -3,2) and D: (x,y,z) = (4,3,6). (c) Show that L and Ly are not parallel lines. (d) Write the parametric equations for L, and L2, and then show that the lines Li and L2 do not...
Given distinct points P1= (x1,y1) and P2= (x2, y2),suppose P=(x,y) is any point on a line through P1 andP2. a. By equating slopes, show that x and y satisfy the equation b. Explain why the equation found in (a) is the equation of a straight line. c. What happens if x2 = x1? **PLEASE SHOW ALL WORK!!
Fourth Homework (1) Let P-(**.0) and Q ( . (a) Find the pole of the line PQ (b) Find the parametrization of the line PQ (c) Does (ch,顽週lie on the line PQ? 克,2 7, ) lie on the line PQ? (2) Find the distance between the lines (1,0,-1) + t(2,3,0) and m (2,-1,3) +s(0, 1,2). (3) Let A and B be two distinct points of S2. Show that X e I d(X, A) = d(X, b)) is a line and...
A topological space X has the Hausdorff property if cach pair of distinct points can be topologically scparated: If x, y E X and y, there exist two disjoint open sets U and U, with E U and y E U and UnU = Ø. (a) Show that each singleton set z} in a Hausdorff space is closed A function from N to a space X is a sequence n > xj in X. A sequence in a topological space...