Scalar Product: Algebraic Measure vs Dot Product

① This interactive GeoGebra example visually contrasts the algebraic measure (signed projection length) of a vector with its dot product (scalar product), clarifying the geometric and algebraic interpretations. ② Core mathematical concepts include the dot product, vector projection, the cosine of the angle between vectors, and the notion of signed length along a direction. ③ Construction highlights: using GeoGebra commands such as Vector, Rotate, Angle, OrthogonalLine, Intersect, Segment, Ray, and Textfield, the example builds vectors u and v, constructs the angle β, projects one vector onto the other, and provides interactive input boxes and buttons to dynamically alter parameters and observe changes in the scalar product. ④ Suitable for high school to early university students, particularly in analytic geometry, introductory linear algebra, or pre-calculus courses; also valuable for teachers demonstrating vector operations and for students exploring geometric intuition behind algebraic formulas.