Physics, as an academic discipline, has its mathematical language, and vectors are its indispensable constituent, even at the pre-mandatory stage. This study aimed to obtain a clear and deep understanding of the perception of students' cognition about the facets of the vectors that form the basis of Newtonian mechanics. This study not only provides fundamental knowledge about the cognition of theoretical vector constructs among students, but also helps to improve existing pedagogies and, eventually, leading to students' discernment. The author developed a Test of Knowledge of Elementary Vector Concepts (TKEVC) that was delivered to engineering students in the first semester who were starting the required engineering physics course during the 1st week of class, before any instruction on vectors. TKEVC was administered to a sample of 476 students from 7 different engineering colleges and universities in India. The result obtained with this test states that only 37% of students enrolled in the engineering physics course have extensive knowledge of vectors to advance physics topics, specifically mechanics, while almost 67% of students enter the course without a complete understanding of the vector principles in any aspect. These findings have substantial implications for math and physics classes. The researcher discussed the questions asked to the students and their answers, emphasizing the conceptual and methodological errors they made. The results have substantial implications for the teaching of engineering mathematics and physics, the professional development of teachers, and the preparation of teams of teachers for engineering institutions.