Using Hiriart-Urruty's signed distance function, we present new definitions of strong slopes for a vector-valued map recently introduced in [E.M. Bednarczuk, A.Y., Kruger, Error bounds for vector-valued functions on metric spaces. Vietnam J. Math. 40 (2012), no. 2-3, 165-180]. With the new presentation, we are able to show that these slopes enjoy most properties of the ones of a scalar function and to obtain exact formula or estimate for them in the convex, strictly differentiable or linear cases. As applications, they are used in the study of error bounds (in particular, a Hoffman-type error bound for a system of linear inequalities in infinite-dimensional space settings), vector optimization (Pareto minima, weak sharp Pareto minima) and calmness of vector-valued maps.