Let KK be a field and, a=((a_1),(a_2),(a_3)),b=((b_1),(b_2),(b_3)),c=((c_1),(c_2),(c_3)),d=((d_1),(d_2),(d_3)) in bbbK^3 Show that a,b,c,d are in an affine plane if and only if det((a_1,b_1,c_1,d_1),(a_2,b_2,c_2,d_3),(a_3,b_3,c_3,d_3),(1,1,1,1))=0 How can I show this?

Patricia Bean 2022-07-20 Answered
Show that vectors are in an affine plane if and only if det=0
Let K be a field and, a = ( a 1 a 2 a 3 ) , b = ( b 1 b 2 b 3 ) , c = ( c 1 c 2 c 3 ) , d = ( d 1 d 2 d 3 ) K 3
Show that a,b,c,d are in an affine plane if and only if
d e t ( a 1 b 1 c 1 d 1 a 2 b 2 c 2 d 2 a 3 b 3 c 3 d 3 1 1 1 1 ) = 0
How can I show this?
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Answers (1)

eri1ti0m
Answered 2022-07-21 Author has 11 answers
The determinent is det ( b 1 a 1 b 2 a 2 b 3 a 3 ) = 0. Then the linear subspace U = span ( b a , c a , d a ) has rank 0,1 or 2.
I think affine plane is defined to be p + V = { p + v v V }, where the point p is in affine space and V is a linear vector (sub)space. So if rank ( U ) = 2, then a+U is a two dimension affine plane and a , b , c , d a + U. If rank ( U ) 1, a+U is a affine line or point, which can contain in a plane that pass the origin (0,0,0).
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