Let a , b , c , d ∈ R 3 and d i m ( s p a n ( a − d , b − d , c − d ) ) ≤ 2 (i.e. the vectors a − d, b − d, c − d are linearly dependant). Prove that a , b , c , d lie on an affine plane.

Question

Let   a  ,  b  ,  c  ,  d  ∈            R        3   and   d  i  m  (  s  p  a  n  (  a  −  d  ,  b  −  d  ,  c  −  d  )  )  ≤  2 (i.e. the vectors   a  −  d,   b  −  d,   c  −  d are linearly dependant). Prove that   a  ,  b  ,  c  ,  d lie on an affine plane.

Preston Espinoza · Accepted Answer

The vectors a−d,b−d,c−d being linearly dependent, one of them, say a−d for example, is a linear combination of the two others:  a  −  d  =  λ  (  b  −  d  )  +  μ  (  c  −  d  )for some   λ  ,  μ  ∈      R  . Hence  a  =  d  +  λ  (  b  −  d  )  +  μ  (  c  −  d  )  ∈  d  +  span  (  b  −  d  ,  c  −  d  )which is an affine subspace of                     R              3  . This subspace contains, in addition to a, also d (if   λ  =  μ  =  0), b (if   λ  =  1  ,  μ  =  0  )) , and c (if   λ  =  0  ,  μ  =  1).Note that the affine subspace above is a plane if b−d,c−d are linearly independent; or a straight line if they are linearly dependent and not both zero (in which case, however, there are infinite planes containing a,b,c,d); or a single point if b−d=c−d=0, since in the latter case we have a=b=c=d.

Let a,b,c,d in RR^3 and dim(span(a−d,b−d,c−d)) <= 2 (i.e. the vectors a−d, b−d, c−d are linearly dependant). Prove that a,b,c,d lie on an affine plane.

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