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The objective of this paper is to provide a holistic summary of ongoing research related to the development, implementation, assessment, and continuous refinement of an augmented reality (AR) app known as Vectors in Space. This Unity-based app was created by the authors and provides a self-guided learning experience for students to learn about fundamental vector concepts routinely encountered in undergraduate physics and engineering mechanics courses. Vectors are a fundamental tool in mechanics courses as they allow for the precise and comprehensive description of physical phenomena such as forces, moments, and motion. In early engineering coursework, students often perceive vectors as an abstract mathematical concept that requires spatial visualization skills in three dimensions (3D). The app aims to allow students to build these tacit skills while simultaneously allowing them to learn fundamental vector concepts that will be necessary in subsequent coursework. Three self-paced, guided learning activities systematically address concepts that include: (a) Cartesian components of vectors, (b) unit vectors and directional angles, (c) addition, (d) subtraction, (e) cross product using the right-hand rule, (f) angle between vectors using the dot product, and (g) vector projections using the dot product. The authors first discuss the app's scaffolding approach with special attention given to the incorporation of Mayer's principles of multimedia learning as well as the use of animations. The authors' approach to develop the associated statics learning activities, practical aspects of implementation, and lessons learned are shared. The effectiveness of the activities is assessed by applying analysis of covariance (ANCOVA) to pre- and post-activity assessment scores for control and treatment groups. Though the sample sizes are relatively small (less than 50 students), the results demonstrate that AR had a positive impact on student learning of the dot product and its applications. Larger sample sizes and refinements to the test instruments will be necessary in the future to draw robust conclusions regarding the other vector topics and operations. Qualitative feedback from student focus groups conducted with undergraduate engineering students identified the app's strengths as well as potential areas of improvement.
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This content will become publicly available on May 8, 2026
Vector dot product: Enhancing understanding and motivation through immersive augmented reality (AR)
Students often struggle to understand the vector dot product, which is a foundational operation used in mathematics and engineering. To improve undergraduate engineering students’ understanding of the dot product, we developed and tested the effects of an augmented reality (AR) app. The app utilized scaffolding and storyline narration to cover: (1) computation of the angle between vectors, and (2) the projection of a force vector onto a line. Students were randomly assigned to either a treatment group to utilize the AR, or a control group for traditional peer collaboration. Pre/post testing was conducted using a 14-item, 100-point test. 61 pairs of pre/posttest data (ARn = 25, controln = 36) were analyzed using ANCOVA. The 20.9-point improvement in the AR group's mean test scores was significantly larger than the 9.33-point increase in the control group. The effect size (partialη2 = 0.135) was considered medium to large. The Instructional Materials Motivation Survey assessed motivation from 12 students in each group. Motivation of the AR group was 19.3% larger than that of the control. The difference was significant with a large effect size. The results suggest that the 3D visualization and immersive qualities of AR may improve learning of vector operations in STEM disciplines.
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- PAR ID:
- 10648530
- Publisher / Repository:
- Sage
- Date Published:
- Journal Name:
- International Journal of Mechanical Engineering Education
- ISSN:
- 0306-4190
- Subject(s) / Keyword(s):
- Augmented reality, dot product, force, motivation, projection, statics, spatial visualization, vector
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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