Vector spaces

In Physics we learn that a force applied at a point O has both magnitude and direction. It is represented by an arrow OP as in Figure 1.1.1, where the length OP represents the magnitude and O to P the direction of the force. If we now apply another force OQ at the point O, the resultant (also called the sum) of the two forces is obtained by the parallelogram law: it is OR where OPRQ is a parallelogram. Also, if the strength of the force OP is doubled without changing the direction, the new force is OS where S is the point on the line OP such that OS = 2 OP. If the direction of the force OP is reversed without altering the magnitude, the new force is OT where T is the point on OP such that OT = −OP with the usual convention. In general, α times the force OP is OW where W is a point on OP (extended either way, if necessary) such that OW/OP = α, where a may be positive, negative or zero.