Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming process. In order to study the role of the mechanical and chemical forces in both of these stages of blood vessel formation, we present a mathematical model which assumes that (i) cells exert traction forces onto the extracellular matrix, (ii) the matrix behaves as a linear viscoelastic material, (iii) the cells move along gradients of exogenously supplied chemical stimuli (chemotaxis) and (iv) these stimuli diffuse or are uptaken by the cells. We study the equations numerically, present an appropriate finite difference scheme and simulate the formation of vascular networks in a plane. Our results compare very well with experimental observations and suggest that spontaneous formation of networks can be explained via a purely mechanical interaction between cells and the extracellular matrix. We find that chemotaxis alone is not a sufficient force to stimulate formation of pattern. Moreover, during vessel sprouting, we find that mechanical forces can help in the formation of well defined vascular structures.