Solving a Vector Space Problem: (a,b,1) Not a Vector Space

[ercagpince]

[SOLVED] a simple vector space problem

Homework Statement

Consider the set of all entities of the form (a,b,c) where the entries are real numbers . Addition and scalar multiplication are defined as follows :
(a,b,c) + (d,e,f) = (a+d,b+e,c+f)
z*(a,b,c) = (za,zb,zc)

Show that vectors of the form (a,b,1) do not form a vector space .

Homework Equations

all equations defining a vector space

The Attempt at a Solution

I managed to find the inverse under addition vector and also the null vector for that vector space , however , I couldn't find any logical explanation or proof why a vector like (a,b,1) do not form a vector space .

 

[dynamicsolo]

Show that vectors of the form (a,b,1) do not form a vector space .

What is the z-component of the resultant vector if you add two of these? Will it still belong to that set of vectors?