3×AB

3×AB is the scalar multiple of the vector AB by the scalar 3. AB typically denotes the directed line segment from point A to point B, viewed as the vector from A to B. Multiplying by 3 scales its magnitude by 3 while preserving its direction. In coordinates, if A and B have position vectors a and b, then AB = b − a and 3AB = 3(b − a). If one takes A as the starting point, the endpoint of the scaled segment is A + 3AB = a + 3(b − a) = 3b − 2a.

Geometrically, 3AB represents a dilation of factor 3 about A along the line AB: the endpoint lies on the ray AB, at a distance three times as far from A as B is (AC = 3·AB, with C on the same line through AB). In vector form this is the point A plus three times the original displacement from A to B.

In two or three dimensions, the components of 3AB are 3(x_B − x_A), 3(y_B − y_A), and, if applicable, 3(z_B − z_A). The concept is fundamental in vector algebra and geometric transformations, including homothety and similarity constructions, where scaling a side or a vector preserves direction but changes length by the scalar factor.

Example: A(1,2) and B(4,6) yield AB = (3,4); 3AB = (9,12). The scaled endpoint from A is (1,2) + (9,12) = (10,14). Note that AB can denote either the vector or the segment; 3×AB is the corresponding scaled vector, while 3×|AB| would be a length.