Thread: Is this a SubSpace?

S is the set of all points P such that vector AP is orthoganol to vector u = (-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:
Let a = AP #1
Let b = 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:
Let a = some arbitrary AP

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

doesnt contain the origin.