Linear Combination of Plane Vectors

This GeoGebra example demonstrates the concept of linear combinations of plane vectors OA and OB. By constructing geometric objects including points, vectors, lines, segments, and polygons, it visually illustrates how any vector in a plane can be expressed as a linear combination of two basis vectors, namely a times OA plus b times OB. The applet includes geometric visualizations of vector addition and scalar multiplication, allowing students to see how scaling a vector and adding vectors together produces new resultant vectors. Interactive sliders enable users to adjust the coefficients a and b in real time, observing how the terminal point of the linear combination moves across the plane. Additional constructions such as intersection points and polygonal regions help clarify the parallelogram law of vector addition and the span of the two basis vectors. The applet is designed for high school mathematics instruction, particularly for Grade 11 or Grade 12 courses covering plane vectors. It supports the understanding of vector decomposition, the geometric meaning of linear combinations, and the foundational ideas of vector spaces. Teachers can use this applet as a classroom demonstration tool, an exploratory activity for students, or a self-study resource. Through dynamic manipulation, learners develop intuition about how two non-collinear basis vectors can generate every vector in the plane, reinforcing the key theorem that any plane vector can be uniquely written as a linear combination of two given non-parallel basis vectors.