Visualizing the Span of Three Vectors

This GeoGebra example demonstrates how the span of three vectors in 3D space is generated by their linear combinations, making the underlying direction space and generative structure visible. The key math ideas are linear combinations, span (the set of all vectors you can form), linear generation of a subspace, and how vector-space concepts relate to geometric representations. Construction-wise, the model creates vector objects u, v, and w using GeoGebra’s Vector command, then forms spanvec as a representative combination of u, v, w with coefficients c1, c2, c3. The construction summary shows the command “Vector” and the involved objects u, v, w, c1, c2, c3, and spanvec, where the coefficients control different combinations and thus illustrate the span behavior. Suitable for high school advanced vector topics and early undergraduate linear algebra, as well as classroom demos and self-study.