Space coordinates, distance, midpoint and the first vectors

Some quantities have a size only, like a mass of 5 kg. Others need a direction too: a force, a velocity, a displacement. We draw them as arrows and call them vectors (in Arabic متجهات). In the plane, “3 m east and 4 m north” is the vector (3, 4); its length is √(9 + 16) = 5 m. In space we use three components. The position vector of a point A(Aₓ, Aᵧ, A_z) is the arrow from O to A, written A = (Aₓ, Aᵧ, A_z). The numbers are its components along the three axes. Its length is the norm ‖A‖ = √(Aₓ² + Aᵧ² + A_z²), which is just the distance OA. Two vectors are added component by component: (a, b, c) + (d, e, f) = (a + d, b + e, c + f). In a picture, place the second arrow at the tip of the first (or complete the parallelogram, as in the plane picture) and the sum goes from the start to the end. Addition behaves like addition of numbers: A + B is again a vector of space, A + B = B + A, (A + B) + C = A + (B + C), the zero vector (0, 0, 0) changes nothing, and every A has an opposite −A = (−Aₓ, −Aᵧ, −A_z) with A + (−A) = (0, 0, 0). In general ‖A + B‖ ≤ ‖A‖ + ‖B‖.