Span of 1 or 2 Vectors in 3D
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This GeoGebra example visualizes the linear span of one or two vectors in three-dimensional space. A single vector spans a line through the origin, while two non-collinear vectors span a plane through the origin. The core mathematical concept is the linear span, the set of all linear combinations of given vectors, illustrating linear dependence and independence. Construction uses GeoGebra commands Line and Plane, with objects such as points A, B, O, P and vectors a, b, f, g, allowing dynamic adjustment of vector directions to observe how the spanned subspace changes. Suitable for senior high school or introductory university linear algebra, ideal for classroom demonstrations of vector spaces and linear combinations.
GeoGebra Commands & Parameters
Line(O, A) → fLine(O, B) → gPlane(O, A, B) → PLine(C, f) → hLine(C, g) → iLine(A, A + B) → jLine(B, A + B) → kLine((-A), B - A) → lLine((-B), A - B) → m
AI Drawing Prompt Reference
In 3D space, center at the origin, draw two vectors from the origin to two movable points. Create a line through the origin and the first point to show the span of the first vector, and another line through the origin and the second point for the second vector. Construct a plane passing through the origin and both points to represent the plane spanned by the two vectors. Additionally, from the first point, draw a line parallel to the second vector; from the second point, draw a line parallel to the first vector. Connect the first point to the sum of the two points, and the second point to that sum, forming a parallelogram. Finally, from the negative of the first point, draw a line to the difference of the second and first points; from the negative of the second point, draw a line to the difference of the first and second points, illustrating vector operations geometrically.
The prompt above was compiled from the original drawing commands and is for reference only.