Can the Resultant of Cross Products of Vectors with Zero Resultant also be Zero?

[STAii]

Greetings !
I have made a conclusion while investigating some vector-related problem.
I am currently trying to proove it (i am sure that if it is true, lot of you would be able to proove it, but i prefer to try that myself for the moment).
What i need is only to know whether or not my conclusion is right.
Here it is :
"If you have a number of vectors (say n) that have a resultant of Zero, and you cross each one of them with a non-zero vector, then find the resultant of the cross results, then it will be zero too"
If this is not really clear, i will try to make it clearer.
Thanks.

 

[HallsofIvy]

In other words, the distributive law holds for cross product:

u x (v+ w)= u x v+ u x w. In the special case that v+ w= 0,

u x v+ u x w= u x 0= 0.

 

[STAii]

Thanks HallsofIvy.
I think i figured out how to proove it even without using distributive law (actually, i didn't know that it holds for cross product).
I will make sure i am not wrong, make everything 'mathematically beautiful' then put it here.