Vectors
This online tool is designed to meet the needs of students and teachers for A-level Vectors. Concepts are carefully developed and presented with the help of diagrams. Examples are provided, as well as questions to reinforce understanding. Interactive ICT tools, using Geogebra, are tailor-made and integrated for the Syllabus 9709. The interactive tools also provide visual aspect of vectors in 3D and promote exploration and independent learning. Worked out solutions of previous examination questions are also presented together with their display in 3D. The tool is divided into three parts: 3D vectors for P1 followed by Lines and Planes for P3.
Vectors in real life
Vectors is widely used across the globe, in particular in aeronautics, navigation (air and sea) and kinematics (linear or circular) among others. Vectors are even used at experimental level in engineering laboratories to model real-life problems involving magnitude and direction. The application of the concept of addition of vectors are often seen during crosswinds when aircrafts approach a runway at an angle, a phenomenon that can be explained using the Triangle Law of vectors. Further, air controllers use vectors together with relative velocity to prevent aircrafts from colliding with each other in busy airspaces, that is, each aircraft is instructed to maintain a specific heading (direction) and an exact airspeed (magnitude). Two dimensional position vectors is another concept that can be superimposed on real-life activities for instance when planning a navigation route and playing billiards where a precise aim and an amount of force are required. The application of vector addition and scalar multiplication can be observed in harbours where two or more tugboats of different pulling power are towing sea cruises and barges in and out of the docking stations.
At secondary level, two dimensional and three dimensional vectors are key components of the A-level mathematics syllabus and they aim at providing learners an authentic connection between real life spatial occurrences and mathematical concepts. The various concepts associated with vectors taught in schools serve as the building blocks of spatial reasoning and at the same time equipping learners with skills of problem-solving, critical-thinking and more important visualization. For instance, some concepts in vectors are subsets of other topics such as translations, relative velocity, complex numbers, and resolution of concurrent forces which include Lami’s Theorem, Triangle Law of Vectors and Parallelogram Law of Vectors. It is therefore imperative for students to have a sound conceptual understanding of vectors should they opt for a career in engineering.
Using ICT as a tool
The National Curriculum lays much emphasis on the use of digital technology as a pedagogical tool across the curriculum to improve understanding and performance. Today many secondary schools are equipped with laptops and projectors, and some educators have already embraced this opportunity to innovate in their teaching. Over the last decade, numerous mathematics software packages have been designed with the intended purpose of facilitating mathematical modelling in schools. One of these tools is Geogebra and it is freely available on the net. The software is user friendly as it is pre-loaded with a graphic user interface (GUI) that allows human-to-machine interaction via visual indicators and graphical icons instead of the command line interfaces. Regarding the teaching of vectors, the Geogebra software helps in translating abstract content into meaningful graphical representation as it is equipped with two dimensional and three dimensional dynamic graphing imagery. As such, 3D objects, lines and planes can be displayed on 3D graphs.