Prove that a projective plane (P,L,I) has the property that every line contains at least three points.

Q: In Cartesian space, the direction of the line (x, y, z) = (a, b, c) +r(d, e, f) is represented by…

A: The given line is (x,y,z)=(a,b,c)+r(d,e,f) So, the point (a,b,c) is on the line and (d,e,f) is a…

Q: Consider the set of vectors {V1, V2, V3} = = {A·A·A) following gives a geometric interpretation of…

A: The set of vectors is given by v1, v2, v3=101, 30-1, 011. We have to find the geometric…

Q: Let A = (0,0,0), B= (1,0,2), C = (2,1,1). Which of the following lines is perpendicular to the plane…

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Q: A plane is perpendicular to [x, y, z] = [3, -30, 24] + s[2, 4, -2] and contains the point…

A: Step 1:​Plane is perpendicular to [x,y,z]=[3,−30,24]+s[2,4,−2]Direction of Line…

Q: Let Z= (-1,2,2) be a point in R³. Find the closest point to Z that lies on the plane given by…

A: We have to find the closest point of  that lie on the plane

Q: Show that the lines L₁ : x = 1 -t, y = 2t, z = 5 + t and L₂: x=1+3t, y = 3-6t, z = 5-3t are parallel…

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Q: Prove incidence theorem 4: Given two intersecting lines, there is exactly one plane containing both

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Q: Find the distance from y to the plane in R spanned by the set {u1, uz} 4 - 3 -3 y = -9 -4 Uz 2 4 1…

A: Solution   Given,       Find the distance from  γ to the plane  in R3       spanned  by the  set…

Q: 6. Find a plane with the P (3, – 4, 5) and contains the line L:= =*

A: It is given that plane passes through (3,-4,5) and the DR's of a given line is 2,-3,5.

Q: Write a scalar equation of each plane. a) [x, y, z] = [2, -5, 4] + s[2, -3, −2] + t[7, -3, 5] b) [x,…

A: Since you have posted a question with multiple subparts, so we will solve only the first three…

Q: Suggest a "vector proof" of the fact that, inR2, two lines with slopes m1and m2 are perpendicular if…

A: Given: ℝ2 To Prove that : m1 m2=-1

Q: is the distance from the point E to the plane N is a vector equation for the line where M and N…

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Q: Let a and b be real numbers. Let L be the line defined as L: 3- 3t, y =-5+ t, z= a+ bt, teR Let D be…

A: Given :- parametric equation of a line L and equation of a plane D are given. Also L and D are…

Q: Let A = (0,0,0), B= (1,0,2), C = (2,1,1). Which of the following lines is perpendicular to the plane…

A: topic - vectors

Q: 2- A plane is defined by any three points that are in the plane. If a plane contains the points…

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Q: Describe each line. a) [x, y] = [2, -5] + t[0, 1] b) [x, y] = [3, 7] + [2, 0] c) [x, y, z] = [-1, 3,…

A: Solution below

Q: Consider the points K(−1, 0, 7) and M(1, −2, 5), the vector MN = , and the line L with symmetric…

A: The dot product of two vectors includes the cosine of the angle between them. When the two vectors…

Q: r = (6,-3,2)+1(2,0,1) + u(1,1,–3)

A: Given point is (0,-1,-8) and the plane is r→=6,-3,2+t2,0,1+u1,1-3

Q: (5, 2, 1) span R³? 1. Do the vectors 1 Explain. [In other words, is the set A = (1, 1, 2), ī2 (2, 3,…

A: We have vectors -  v1 = (1,1,2) , v2 = ( 2,3,4) , v3 = (3,4,6) , v4 = (5,2,1)  We have to check this…

Q: a) Find an equation of the line through P(1,-2,4) with direction vector v=[2,4,-4] b) Find the…

A: Here we are solving part c.

Q: Determine the intercepts of each plane. a) [x, y, z] = [1, 8, 6] + s[1, –12, –12] + t[2, 4, -3]

A: As per policy, only the first question is answered as you have asked multiple questions asked in a…

Q: Let v₁ = (1,3,5), v² = (-2, 1, 7) and let P be the plane spanned by v¹ and v². (That is, P = (v1,…

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