[justify]Many reports have been written regarding the difficulty faced by students when attempting to understand vector concepts, particularly in the context of modeling in physics related problems, see for example Knight (1995), Nguyen & Meltzer (2003) and Barniol & Zavala (2009, 2010, 2012 and 2015). Also, many successful attempts to overcome a wide range of difficulties to understand vectors have been performed in the form of modeling tasks Doerr,H. M. (1996), computer applications (Cataloglu, E., 2006; Tsegaye, K.& Baylie, D. & Dejne, S. 2010), worksheets (Umporn, W. & KarnpitchaB. &  Narumon, E., 2015; Alarcón, H. & Zavala, G., 2006), and many more, nevertheless, the use of [i]dynamic geometry software[/i] has been widely proven to support the learning process of mathematics related subjects (Hohenwarter, M. & Preiner, J. 2007; Stahl, G. 2010; Forsythe, S. 2010), and now, with the release of GeoGebra 5 which includes a 3D view, visualization of 3D vectors is possible. This work focuses on visualizing 3D Cartesian vectors in order to deduce correctly the relationships between the vector components, the magnitude of the vector, and its direction cosines, and it is used as support to a collaborative worksheet on 3D vectors designed by Alarcón,H. & Zavala, G. (2006).[/justify]

[justify]The target learners are first semester students in a private Mexican university who are taking an introductory course focused on mathematical tools applied to physics, with special attention to modeling with calculus and vectors. The mathematical background of the students varies significantly given that they come from many different secondary schools spread all around Mexico and Latin America, but most of them have very low level of basic mathematical tools such as algebra and trigonometry, although the only requisite to take the subject is basic algebra.[br]The course was designed in flipped classroom format, in the sense that the students are assigned to watch videos in a MOOC specially designed for the subject (Fernández de Lara, R., Salmón, R. & Zavala, G., 2013), these videos must be watched and a selfdiagnostic must be solved previous to the first session of each topic during the semester; the students are also encouraged to read the course text book by Alarcón, H. & Zavala, G. (2008). [br]During the first session of each topic the students solve a worksheet collaboratively, these worksheets were also specifically designed for the course by Alarcón, H. & Zavala, G. (2006). During the solution of the worksheet, the students are frequently aided by the instructor and a teaching assistant by giving them feedback and suggesting ways to reason about the solution of the sequenced questions and problems.[br]The applet designed for this work deals with 3D vectors, so it is important to mention that by this point of the course the students learned previously about right triangle trigonometry, graphical[br]representation of vectors, calculation of rectangular components in 2D vectors, and basic vector operations such as addition, subtraction and multiplication by a scalar.[/justify]

[justify]The applet was tested on two groups of students during the semesters of August-December 2014 and January-May 2015, each group in the first semester had 60 students while the groups in the second semester had 30 students each, making a total of 180 students.[br]The activity to work collaboratively is about 3D Cartesian vectors containing several objectives to[br]achieve, but the applet focuses only on three of them:[br]a) To deduce the Pythagorean formula to calculate the magnitude of the vector given its components.[br]b) To relate the sign of any vector component with the range of values of the corresponding direction angle.[br]c) To deduce the relation between any component, its corresponding direction angle and the[br]magnitude of the vector, i.e. [math]A_x=|\vec{A}|cos(\alpha)[/math].[br]Although GeoGebra is used during the course for many different topics, students know very basic use of the software, so they are not required to load the applet in their own computer, although the applet was shared to them given that some students want to interact with the 3D vector themselves, but for most of the students the applet was manipulated by the instructor projecting the images in all four screens found in the classroom, prompting them to make the appropriate observations of the vector geometry.[br]Once the applet is loaded and projected onto the screens in the classroom, it is important to describe the perspective from which the 3D system and the vector are seen, the students need to identify all three axes and their orientations in order to distinguish where the vector is located and be aware of its direction, also, any sliding to change vector components or rotation of the image to visualize different perspectives must be done slowly to avoid the students to get confused.[br]For the objectives listed, a) and c) are eased by rotating the image in order to visualize the appropriate right triangles in a vector with all components positive, and for tasks in objective b) the components are changed to zero or negative values gradually in order to observe the possible values of the direction angles; the handling of the applet for each objective is described in the following subsections.[/justify]

[justify]Given that the session for the activity lasts 80 minutes, it was not possible to implement a test in order to have an accurate measure of the understanding of the concepts explored during the activity, and given that the exam to test the related competencies was held weeks after, while many other tasks and learning experiences on vectors were done, the best possible measure of the impact of the applet was the comparison in the performance on the activity with sessions in the previous semesters, specifically, it was noticed that students needed less instruction to complete the objectives listed after the demonstration was shown, and since these are the most difficult tasks, the time to complete the worksheet was greatly reduced, allowing more than 70% of the students to complete the task on time, compared to less than 50% of the students who finished the activity in previous semesters.[/justify]

[justify]GeoGebra 5 with its 3D view offers a great potential to overcome many difficulties faced by students when learning about Cartesian 3D vectors, with the applet presented students can easily explore and discover the formula to calculate the magnitude of a 3D vector, relate the direction angles to the sign of the corresponding component and establish the mathematical relation between the latter. [br]The applet is also used in a further session to show the projection of a vector on any of the coordinate planes, with special attention to the projection on the [math]xy[/math] plane to visualize the polar angle [math]\phi[/math] from spherical and cylindrical coordinate vectors. [br]An improvement to the applet would be the addition of sliders to control the values of the components rather than sliding the points on the axis were the components are drawn, also adding check boxes to hide or show the different relevant objects would allow more students to interact with the applet in their computer rather than having the instructor making the demonstration on the classroom screens.[/justify]

It is suggested to download the applet and load it into the desktop application in order to rotate the view.

[justify]Alarcón, H. & Zavala, G. (2008). [i]Introducción ala física universitaria: Conceptos y herramientas[/i]. Mexico, D. F.: EditorialTrillas.[br][br]Alarcón, H. & Zavala, G. (2006). [i]Introducción ala física universitaria: Manual de actividades[/i]. Mexico, D. F.: Editorial Trillas.[br] [br]Barniol, P., ZavalaG. (2009). Investigation of Students’ Preconceptions and Difficulties with the Vector Direction Concept at a Mexican University. [i]AIP Conference Proceedings[/i], 1179, 85-88.[br] [br]Barniol, P., ZavalaG. (2010). Vector Addition: Effect of the Context and Position of the Vectors. [i]AIP Conference Proceedings[/i], 1289, 73-76.[br] [br]Barniol, P., ZavalaG. (2012). Students’ Difficulties with Unit Vectors and Scalar Multiplication[br]of a Vector. [i]AIP Conference Proceedings[/i],1413, 115-118.[br] [br]Barniol, P., ZavalaG. (2015). Force, Velocity and Work: the Effect of Different Contexts on Students’ Understanding of Vector Problems using Isomorphic Problems. [i]Physical Review Special Topics – PhysicsEducation Research[/i], 11 (1), 020115-1-020115-15.[br] [br]Cataloglu, E. (2006).Open Source Software in Teaching Physics: A Case Study on Vector Algebra and[br][br]Visual Representations. [i]The TurkishOnline Journal of Educational Technology – TOJET[/i] January 2006 ISSN:1303-6521 volume 5 Issue 1 Article 8[br] [br]Doerr, H. M. (1996).Integrating the study of trigonometry, vectors, and force through modeling. [i]SchoolScience And Mathematics[/i], (8), 407.[br] [br]Fernández de Lara,R., Salmón, R., Zavala, G. (2013). MOOC: Conceptos y Herramientas para la[br]Física Universitaria. [url=https://www.coursera.org/course/cyhfisica] https://www.coursera.org/course/cyhfisica[/url][br] [br]Forsythe, S. (2010).   A study of the effectiveness of a Dynamic Geometry Program to Support the Learning of Geometrical Concepts of 2D Shapes. [i]Proceedings of the British Society for Research into Learning Mathematics[/i] 30(2).[br] [br]Hohenwarter, M.,Preiner, Z. (2007). Dynamic Mathematics with GeoGebra. [i]Journal of Online Mathematics and its Applications[/i], MAA ID 1448,Vol. 7. [br] [br]Iskander, W., Curtis,S. (2005). Use of colour and interactive animation in learning 3D vectors. [i]Journal Of Computers In Mathematics And Science Teaching[/i], (2), 149.[br] [br]Knight, R. D. (1995).The Vector Knowledge of Beginning Physics Students. [i]The Physics Teacher[/i], 33 (2), 74-77.[br] [br]Nguyen, N., Meltzer,D. E. (2003). Initial Understanding of Vector Concepts among Students in Introductory Physics Course.  [i]American Journal of Physics[/i], 71 (6),630-638.[br] [br]Stahl, G. (2010). Computer Mediation of Collaborative Mathematical Exploration. ICLS '10 Proceedings of the 9th International Conference  of the Learning Sciences – 2, 30-33.[br][br]Tsegaye, K., Baylie,D., & Dejne, S. (2010). Computer based teaching aid for basic vector operations in higher institution Physics. [i]Latin-American Journal Of Physics Education[/i], [i]4[/i](1), 3.[br] [br]Umporn, W.,Karnpitcha, B., & Narumon, E. (2015). Teaching Basic Vector Concepts: A Worksheet for the Recovery of Students' Vector Understanding. [i]Eurasian Journal Of Physics & Chemistry Education[/i], [i]7[/i](1), 18.[br] [/justify][table] [tr] [td][br] [/td] [/tr] [tr] [td][br][/td] [/tr][/table]