The components of a state vector must add up to 1. If [tex]x_i[/tex] denotes the probability that the system is in state [tex]i[/tex], and given the conditions above, then you have
[tex]\begin{cases}x_3=4x_1\\x_4=0\\x_2=0.2\\x_1+x_2+x_3+x_4=1\end{cases}[/tex]
The last equation reduces to
[tex]x_1+0.2+4x_1+0=1\implies 5x_1=0.8\implies x_1=0.16\implies x_3=4(0.16)=0.64[/tex]
So the state vector is
[tex]\mathbf x=\begin{bmatrix}0.16\\0.2\\0.64\\0\end{bmatrix}[/tex]
