I cannot figure out whether a set is a vector space or not given certain operations.

There are 10 vector space axioms.

1: Ifuandvare objects in V, thenu+vis in V.

2:u+v=v+u

3:u+ (v+w) = (u+v) +w

4: There is an object0in V, called afor V, such thatzero vector0+u=u+0=ufor alluin V.

5: For eachuin V, there is an object -uin V, called aofnegativeu, such thatu+ (-u) = (-u) +u=0.

6: Ifkis any scalar anduis any object in V, thenkuis in V.

7:k(u+v) =ku+kv

8: (k+l)u=ku+lu

9:k(lu) = (kl)u

10: 1u=u

In the exercies I'm to determine which sets are vector spaces under the given operations, and for those that are not I'm to list all the axioms that fail.

An exercise.

The set of all tripples of real numbers (x, y, z) with the operations (x, y, z) + (x', y', z') = (x + x', y + y', z + z') and k(x, y, z) = (0, 0, 0).

According to the answers this is not a vector space since axiom 10 fails.

Am I understanding it correct when I do 1(x, y, z) = (x, y, z) != (0, 0, 0), if either x != 0, y != 0 or z != 0?

How do I test axiom 7? Or 8 and 9 as well. I can't get my head around it.