The midpoint M of points A and B is given by the equation M − A = B − M . Prove that M = 1 2 ( A + B ) using only the vector space axioms.

Question

The midpoint M of points A and B is given by the equation   M  −  A  =  B  −  M . Prove that   M  =      1    2    (  A  +  B  ) using only the vector space axioms.

Zane Barry · Accepted Answer

Step 1A  and B and M are points from the vector space. So, adding M to both side of the equation   M  −  A  =  B  −  M gives,   M  −  A  +  M  =  B  −  M  +  M , or,   2  M  −  A  =  B (as, -M is the inverse of M,   −  M  +  M  =  0 ). Again, adding A to both side you will get   2  M  −  A  +  A  =  B  −  A  +  A , by same argument,   2  M  =  B  +  A . As the associative set is a field in case of vector space, we can take multiplicative inverse of scalar. Hence,       1    2    ⋅  2  M  =      1    2    (  B  +  A  ) , or,  M  =      1    2              (        A  +  B            )      We can write   B  +  A  =  A  +  B , as, addition of vectors is commutative in a vector space.

The midpoint M of points A and B is given by the equation M − A = B

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