I assume this is to be solved numerically.

(S intersect T) is a subspace, W. Thus W=perp(perp(W)).

perp(W)=(perp(S) union perp(T))

To annihilate the last set, a vector must annihilate the basis for perp(S)andthe basis for perp(T). In other words, it must lie in the null space of the matrix containing these (4) vectors. This null space is one dimensional, being generated by the vector

-0.5477

-0.4472

0.2236

0.6708

Obtained from Matlab by the command:

null([null([ 1 1 1 0; 0 -1 1 0]) null([1 0 0 1; 0 2 0 -1])])