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5. Given parallel lines l and m. Given points A and B that lie on the opposite side of m from l; i.e., for any point P...
Q2. Let u and v be non-parallel vectors in Rn and define Suv (a) Does the point r lie on the straight line through q with direction vector p? (b) Does the point s lie on the straight line through...
Q2. Let u and v be non-parallel vectors in Rn and define Suv (a) Does the point r lie on the straight line through q with direction vector p? (b) Does the point s lie on the straight line through q with direction vector p? (c) Prove that the vectors s and p -r are parallel. (d) Find the intersection point of the line {q+λ p | λ E R} and the line through the points u and v. Q3....
6. Prove: Through a point P there are exactly three lines parallel to p, the polar...
6. Prove: Through a point P there are exactly three lines parallel to p, the polar of P (i.e., the three lines have no points in common with line p)
Given two lines in space, either they are parallel, they intersect, or they are skew (lie in para...
The same question but has 3 parts
Given two lines in space, either they are parallel, they intersect, or they are skew (lie in parallel planes). Determine whether the lines below, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. Otherwise, find the distance between the two lines. Select the correct choice below and fill in the answer box(es) to complete your choice. Type exact answers, using radicals as needed.)...
Prove that (P;L; d) not satisfy postulate 6 of neutral geometry L = {1 c R313(a,b,c.),...
Prove that (P;L; d) not satisfy postulate 6 of neutral geometry
L = {1 c R313(a,b,c.), (u, v, w) є R3, such that I = {(a, b, cht.(u, v, w)|t є R)), and d: Px PR U, V, W T22 Postulate 6 (The Plane Separation Postulate). For any line l, the set of all points not on l is the union of two disjoint subsets called the sides ofl. If A and B are distinct points not on t, then...
[7 points) Given the point A(1,2,1) and the vector v = (2,1,5): (a) Find the point...
[7 points) Given the point A(1,2,1) and the vector v = (2,1,5): (a) Find the point B such that AB V. (b) Find the unit vector u in the opposite direction of v. (c) Find a vector equation for the line L which passes through A and is parallel to v. (d) True or False?: The line L is a subspace of R3. Give a brief explanation of your answer. (e) Find a general equation of the plane P that...
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...
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...
Let A, B, C, and D be four distinct points in the plane. Suppose that no three of them lie on a l...
Let A, B, C, and D be four distinct points in the plane. Suppose that no three of them lie on a line and A, C are on opposite sides of the line BD. The lengths of the line segments AB, BC, CD and DA are 1, 2, 3 and 4 respectively. (a) What is the range of possible values for the length x of the line segment BD? You should justify your answer carefully! [5 marks] (b) Now suppose...
using these axioms prove proof number 5 1 - . Axiom 1: There exist at least...
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,...
1 A capacitor is constructed from a pair of thin square plates of side-width /(i.e. Area-P)...
1 A capacitor is constructed from a pair of thin square plates of side-width /(i.e. Area-P) and separation 3d. Let's assume 3d<<I. A solid square conducting plate of thickness d with the same side-width I (i.e. Area - P) is inserted between the plates as shown below Charge +0 is supplied on the top plate and-0 is supplied on the bottom plate. What is the electric field E between the conductors? (Express E by Q.d l and &) a) b)...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1&#...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
Q2. Let u and v be non-parallel vectors in Rn and define Suv (a) Does the point r lie on the straight line through q with direction vector p? (b) Does the point s lie on the straight line through q with direction vector p? (c) Prove that the vectors s and p -r are parallel. (d) Find the intersection point of the line {q+λ p | λ E R} and the line through the points u and v. Q3....
6. Prove: Through a point P there are exactly three lines parallel to p, the polar of P (i.e., the three lines have no points in common with line p)
The same question but has 3 parts
Given two lines in space, either they are parallel, they intersect, or they are skew (lie in parallel planes). Determine whether the lines below, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. Otherwise, find the distance between the two lines. Select the correct choice below and fill in the answer box(es) to complete your choice. Type exact answers, using radicals as needed.)...
Prove that (P;L; d) not satisfy postulate 6 of neutral geometry
L = {1 c R313(a,b,c.), (u, v, w) є R3, such that I = {(a, b, cht.(u, v, w)|t є R)), and d: Px PR U, V, W T22 Postulate 6 (The Plane Separation Postulate). For any line l, the set of all points not on l is the union of two disjoint subsets called the sides ofl. If A and B are distinct points not on t, then...
[7 points) Given the point A(1,2,1) and the vector v = (2,1,5): (a) Find the point B such that AB V. (b) Find the unit vector u in the opposite direction of v. (c) Find a vector equation for the line L which passes through A and is parallel to v. (d) True or False?: The line L is a subspace of R3. Give a brief explanation of your answer. (e) Find a general equation of the plane P that...
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,...
1 A capacitor is constructed from a pair of thin square plates of side-width /(i.e. Area-P) and separation 3d. Let's assume 3d<<I. A solid square conducting plate of thickness d with the same side-width I (i.e. Area - P) is inserted between the plates as shown below Charge +0 is supplied on the top plate and-0 is supplied on the bottom plate. What is the electric field E between the conductors? (Express E by Q.d l and &) a) b)...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
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