When we administered our Test of Understanding of Vectors (TUV) to students who were finishing a physics university remedial course (that covers subjects of a traditional high school physics course), we observed that they have considerable difficulties in calculating the x-component of a vector when the angle given is measured from the y-axis to the vector. As a result of this finding, we decided to design a tutorial worksheet that guides students through the development of a conceptual framework in this subject. The worksheet was implemented with 264 students of the same course in another semester. Upon using the TUV to evaluate the students’ understanding, we confirmed that the tutorial worksheet had facilitated their learning. This tutorial worksheet is presented in the appendix Appendix Tutorial worksheet Calculation of the x- and y-components of a vector 1. Trigonometry of the triangle Consider the triangle shown in the figure. Is it a right triangle? _______ Explain. Complete the next equations: cos(θ) = _______, sin(θ) =_______. 2. Sketch of the x- and y-components of a vector The figure shows vector A. Sketch the x-component of the vector naming it Ax and sketch the y-component of the vector naming it Ay. Notice that the x-component can be sketched below or above the vector. Also notice that the y-component can be sketched to the left or to the right of the vector. Identify and name these two ways of sketching Ax and Ay. In the previous drawings, notice that a triangle is formed above the vector and another triangle is formed below the vector. Are these right triangles? 3. First calculation of the x- and ycomponents of a vector The figure shows vector B with a magnitude of 5 units that forms an angle θ = 30∘ measured from the x-axis. Calculate the x-component of the vector (Hint: use the triangle that is formed below the vector). Using the same triangle, calculate the y-component of vector B. 4. Second calculation of the x- and y-components of a vector The figure shows vector C with a magnitude of 5 units that forms an angle ϕ = 40∘ measured from the y-axis. Given what you have learned in the previous sections, calculate the x-component of the vector (Hint: use the triangle that is formed above the vector). Using the same triangle, calculate the y-component of the vector C. 5. Integrated vision Consider section 3. The angle given was measured from which axis? Which trigonometric function did you use to find the x-component of the vector? Consider section 4. The angle given was measured from which axis? Which trigonometric function did you use to find the x-component of the vector? What can you conclude about the calculation of the x-component of a vector? Following the same reasoning for the calculation of the y-component in sections 3 and 4, what can you conclude about the calculation of the y-component of a vector? In physics courses students are accustomed to repeating the phrase: “To calculate the x-component of a vector, one has to use the cosine function”. Is this statement complete? What should be added to this phrase in order for it to be complete? and might be used by other physics instructors who teach this material in high schools, colleges or universities.
Keywords:
vector components; vector concepts; tutorial worksheet; instructional material
