Show: Vectors e.g.(a,b,1) do not form vector space.

Homework Equations

definition of null vector,
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The Attempt at a Solution

null vector : $|0 \rangle = (0,0,0)$
inverse of (a,b,c) = ( - a, -b, -c)
vector sum of the two vectors of the same form e.g. (c,d,1) + ( e,f,1) = ( c+e, d+f, 2) does not have the same form. So, the vectors of the form ( a,b,1) do not form a vector space.

Is this correct?

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Yes, or even easier $(0,0,0) \notin \{\,(a,b,1)\,\}$.
Yes, or even easier $(0,0,0) \notin \{\,(a,b,1)\,\}$.