Direction Cosines: 3D Vector Decomposition and Geometric Visualization

This GeoGebra example demonstrates the concept of direction cosines through the decomposition of a vector in three-dimensional space. It visualizes how a vector's direction is characterized by the angles α, β, γ it makes with the coordinate axes, and their corresponding cosine values. The construction uses commands such as Intersect to find key points, Angle to measure the direction angles, and Polygon to create auxiliary faces. Objects include vectors v, v_1 through v_6, points O, X, Y, Z, V, A, B, C, and multiple text labels (text1 to text4) along with auxiliary lines (p, q, r, a, b, c, d, e, f). This example is suitable for high school or early college courses in solid geometry and vector algebra, helping learners grasp how direction cosines relate vector components to axis angles, and serving as a visual aid for coordinate geometry and vector calculations.