Finding an orthogonal basis of the subspace spanned by given vectors 
Let W be the subspace spanned by the given vectors. Find a basis for $W^\perp$.
  $$v_1=(2,1,-2) ;v_2=(4,0,1)$$

Well I did the following to find the basis. 

$$(x,y,z)*(2,1,-2)=0$$ $$(x,y,z)*(4,0,1)=0$$

If you simplify this in to a Linear equation

$$2x + y - 2z = 0$$ $$4x + z = 0$$

Now bu placing this in a vector and performing row echelon I get
$$ w = 
\left[ \begin{array}{ccc|c}
1 &\ 0 &\ 1/4 &0\\ 
0&\ 1 &\ -5/2 &0
\end{array} \right]$$
By solving this I get

$$x=-1/4 t$$
  $$y=5/2 t$$
  $$z=t$$

By this I get the basis to be.

$$[-1/4, 5/2 ,1]$$

I don't see that the answer is correct because you get the vector space in the someway. Please tell me if I used the correct method.
Thank you

Since you work in $\mathbb R^3$ so simply take $v_3=v_1\wedge v_2=(1,-10,-4)$.
