Calculating a vector perpendicular to a unit vector

Question

"find unit vector x in direction of w = (2,1,2)" then "find vector perpendicular to x"

so I calculate x as w/length(w) = w/sqrt(2^2 + 1^2 + 2^2) = w/sqrt(9) = w/3

and a perpendicular vector of w = 1/3(2,1,2) could be = 1/3(1,-2,0)

Strangely, the answer key gives the perpendicular as 1/sqrt(5)(1,-2,0)

Is this an error or are my calculation mechanics wrong somewhere?

Answer 1

A vector perpendicular to the given vector $W$ is $X=(x,y,z)$ such that it satisfies

$W.X=0$

which implies values of (x,y,z) can be such that

$2x+y+2z=0$

There are infinitely many vectors which satisfies above equation.

Both, your answer and text book answer is correct since they satisfy the above equation.