Explain why a line in R3 that does not pass through the orig

Explain why a line in R^3 that does not pass through the origin is not a vector space by exhibiting a vector space axiom that is not satisfied.

Solution

A line \'L\' is a set of points

So let V={(x,y,z)/ (x,y,z) is a point on the line L}

To say that V is a vector space (V, +) should be an abelian group

But the identity element is (0,0,0) with respective to \'+\'

Because we are given a line which \'L\' do not passes through origin, (0,0,0) is not a point in V

I.e., V has no identity element and hence can not be a group

Thus V is not a vector space.

Therefore a line that donot pass through the orign can not be a vector space.