Vector Equation of a Line vs Point-Gradient Form

This GeoGebra example visually compares two common representations of a straight line in 2D: the vector equation and the point-gradient (point-slope) form. It demonstrates how both forms describe the same geometric line using a direction vector and a point, or a slope and a point. Core mathematical concepts include direction vectors, position vectors, slope, and parametric representation of lines. The construction uses GeoGebra commands such as Vector to create vector v, Line to draw the line, and Slope to compute its gradient. Objects include points B and A, vectors v and v_1, lines a, a_1, and g, along with text labels text1 through text5 to display the equations. This is suitable for high school or early college students learning analytic geometry, helping them see the equivalence between algebraic forms and their geometric meaning. By dragging points or adjusting the vector, learners can observe how both equations update simultaneously, reinforcing the connection between direction, slope, and position.