Scalar multiples, unit vectors, i j k, the vector AB and direction cosines

Three special unit vectors point along the axes: i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1) (in Arabic متجهات الوحدة الأساسية). Any vector splits into three parts along the axes: (Aₓ, Aᵧ, A_z) = Aₓ(1, 0, 0) + Aᵧ(0, 1, 0) + A_z(0, 0, 1) = Aₓi + Aᵧj + A_zk. In the picture the diagonal of the box is 2i + 3j + 4k: go 2 along x, 3 along y, 4 along z. All the rules for components hold with i, j, k: add, subtract and multiply the coefficients of i, of j and of k. The vector from a point A to a point B is found by going from A back to the origin and then on to B: AB = OB − OA = B − A (end minus start). The reverse vector is BA = −AB, and the distance AB is the norm ‖AB‖.