S is the set of all points P such that vectorAPis orthoganol to vectoru= (-3,2,-3), where point A = (1,-2,3)

Is S a subspace of R^3 ?

---- The answer is "NO" but I dont know why.... can someone please explain?

------- My Work, which is wrong...

I tried to solve it symbolically by saying S = {(p1,p2,p3) |AP.u= 0 }

and then checking if it was closed under vector Addition & scalar multiplication.

Addition:

Leta=AP#1

Letb=AP#2

We know

a.u= 0

b.u= 0

So under addition

(a+b).u= 0

a.u+b.u= 0

(0) + (0) = 0

Therefore closed under addition.

Multiplication:

Leta= some arbitraryAP

We know

a.u= 0

Then

b= k.a, where k = some scalar

b.u= 0

(k.a).u= 0

k.(a.u) = 0

k.(0) = 0

Therefore closed under scalar multiplication. Therefore it is a subspace.

-------

The above work is wrong. I'm not sure why. Can someone please explain? Thanks