Need help understanding Real Vector Spaces

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

There are 10 vector space axioms.

1: If **u** and **v** are objects in V, then **u** + **v** is in V.

2: **u** + **v** = **v** + **u**

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

4: There is an object **0** in V, called a **zero vector** for V, such that **0** + **u** = **u** + **0** = **u** for all **u** in V.

5: For each **u** in V, there is an object -**u** in V, called a **negative** of **u**, such that **u** + (-**u**) = (-**u**) + **u** = **0**.

6: If* k* is any scalar and **u** is any object in V, then *k***u** is in V.

7: *k*(**u** + **v**) = *k***u** + *k***v**

8: (*k* + *l*)**u** = *k***u** + *l***u**

9: *k*(*l***u**) = (*kl*)**u**

10: 1**u** = **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.