<p>The Laplacian of a vector (aka vector Laplacian) is generally derived in orthogonal curvilinear coordinates through an elaborated process that is based on the classical vector definition of the difference between the grad div and the curl curl of any arbitrary vector field. In this article a totally different matrix notation and formalism will be introduced departing from classical vector and tensor procedures and techniques, firstly to obtain a final set of results based initially on the above-mentioned conventional scientific path. However, and most importantly, a second alternative solution of the problem relying on an entirely independent approach, also employing matrix representation and calculus, will be introduced. This new alternate option certainly could contribute to expanding the versatility of this innovative matrix methodology in facilitating the comprehension for prospective advanced students who are interested in grasping this rather intricate theory encompassing orthogonal curvilinear coordinate applications. Finally, one last way to determine the vector Laplacian operating on a vector defined as a function of the unit (basis) local vectors will be shown also using a matrix procedure. Recall that the vector Laplacian operator boasts definite relevance in the theoretical arsenal of mathematical physics applicable to various scientific disciplines, including geodesy. This assertion will be corroborated with one example related to the general theory of elastodynamics which is inserted herein as a practical illustration.</p>