On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5

Mayada Abualhomos¹, Ahmed Atallah Alsaraireh², Malik Bataineh³, Nabeela Abu-Alkishik⁴, Isra Al-Shbeil⁵, Abdallah Al-Husban^6,*

¹Applied Science Private University Amman,11931, Jordan

²Department of Computer Information Systems, The University of Jordan – Aqaba, Jordan

³Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan,

⁴Department of Mathematics, Jerash University, Jerash, Jordan

⁵Department of Mathematics, Faculty of Sciences, University of Jordan, Amman 11942, Jordan

⁶Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan

Emails: abuhomos@asu.edu.jo; a.alsarairah@ju.edu.jo; msbataineh@just.edu.jo; Nabeelakishik@yahoo.com; i.shbeil@ju.edu.jo; dralhosban@inu.edu.jo

Abstract

This paper will study the necessary conditions and sufficient conditions for the problem of orthogonality in 4-plithogenic and 5-plithogenic vector spaces, where we provided a definition of the real scalar products defined above these spaces, and we used the new concept of scalar product that was defined in determining the necessary and sufficient condition for the orthogonality of symbolic 4-plithogenic and 5-plithogenic vectors, in addition to calculating various norms and their metric properties. We have also provided some examples that aim to explain the scientific contribution of this work.

Keywords: plithogenic vector space; inner product; plithogenic norm; orthogonal basis; ortho-normed vectors.