Let Φ be a set of isometries of a space A. Three distinct points a, b, c ∈ A we call “Φ-collinear” if there is a non-trivial (i.e., not the identity) isometry T ∈ Φ exists for which a, b, c are fixed points (that is, T(a) = a, T(b) = b, T(c) = c). For the following cases, answer the following question and prove your answers: Are three different points are collinear (in the usual sense of the space in question) then and only if they are Φ-collinear? (a) A = E2 and Φ = all isometries of E2 (b) A = E2 and Φ = all direct isometries of E2 (c) A = E3 and Φ = all isometries of E3 (d) A = E3 and Φ = all direct isometries of E3 (e) A = En and Φ = all isometries of En, where n ≥ 4. (f) A = En and Φ = all direct isometries of En, where n ≥ 4. The E's are supposed to be double struck but it wont show like that so Im mentioning it here. please if able provide some explanation with the taken steps. Thank you in advance.
Let Φ be a set of isometries of a space A. Three distinct points a, b, c ∈ A
we call “Φ-collinear” if there is a non-trivial (i.e., not the identity) isometry
T ∈ Φ exists for which a, b, c are fixed points (that is, T(a) = a, T(b) = b,
T(c) = c).
For the following cases, answer the following question and prove your answers: Are
three different points are collinear (in the usual sense of the space in question) then and
only if they are Φ-collinear?
(a) A = E2 and Φ = all isometries of E2
(b) A = E2 and Φ = all direct isometries of E2
(c) A = E3 and Φ = all isometries of E3
(d) A = E3 and Φ = all direct isometries of E3
(e) A = En and Φ = all isometries of En, where n ≥ 4.
(f) A = En and Φ = all direct isometries of En, where n ≥ 4.
The E's are supposed to be double struck but it wont show like that so Im mentioning it here.
please if able provide some explanation with the taken steps. Thank you in advance.
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