This paper introduces and characterizes a new family of continuous probability distributions applicable to norm distributions in three-dimensional random spaces, specifically for the Euclidean norm of three random Gaussian variables with non-zero means. The distribution is specified over the semi-infinite range $[0,\infty)$ and is notable for its computational tractability. Building on this foundation, we also introduce a separate family of continuous probability distributions suitable for power distributions in three-dimensional random spaces. Despite being previously unknown, these distributions are attractive for numerous applications, some of which are discussed in this work.