Logical vectors in MATLAB compare elements of vectors and return a logical vector with the same size where each element indicates whether the comparison is true (1) or false (0). Logical vectors can be used to create discontinuous graphs by multiplying a vector by a logical vector that filters out negative values. They can also avoid division by zero errors by replacing zeros with a small nonzero value like eps. Common logical functions for vectors include any(), which returns 1 if any element is nonzero, and all(), which returns 1 only if all elements are nonzero.