Could anyone explain why:

All vectors of the form (a, 0, 0) is closed under addition and multiplication,

(a, 0, 0) + (b, 0, 0) = (a + b, 0, 0)

k (a, 0, 0) = (ka, 0, 0)

While something like:

All vectors of the form (a, 1, 1)

Is not closed under addition or multiplication?

(a, 1, 1 ) + (b, 1, 1) = (a + b, 2 ,2)

Look! Is the vector $\displaystyle (a+b,2,2)$ of the same form as $\displaystyle (a,1,1)$ ? Of course not since the later has 1 in its 2nd and 3rd coordinates!...and there you've shown why this set isn't closed under sum (and something very similar shows that it isn't closed under multiplicatiom, either). Tonio
k( a, 1, 1) = (ka, k, k)

I worked it out, but I don't know why this is not closed under addition or multiplication.

Any help is appreciated!