The latest VTOL (vertical take-off and landing) aircraft are described in the press as having ‘vectored thrust’. The jet can indeed be rotated relative to the aircraft so as to produce an orientated propelling force. In less newsworthy applications, physicists and engineers have been dealing with vectored thrust for a century or so, and in an immense variety of situations.

But it is not the only force that is vectored. The term vector may refer to any directed quantity, the most elementary instance being the geometrical displacement in the previous chapter. When dealing with vectors, ordinary numbers are called scalars. In physics all vector quantities are ultimately related to displacements, so it is both convenient and sufficient to single them out for discussion. The displacement (x, y, z) from the origin, or from any point, has a magnitude and a direction. Attending first to the x-direction (the axes, remember, are arbitrary, but have usually been chosen to simplify some geometry), then the displacement (1,0,0) is called the unit vector in the x-direction. It is denoted by the single bold-face symbol i. Positive or negative multiples of i can be added in any order to give a larger or smaller displacement along the x-axis. The rule is: multiplying a vector by a scalar changes its magnitude by the same factor and multiplying by – 1 reverses its direction (see fig. 4.1).