Mathematical Methods for Physics and Engineering - Matematica.NET

7.3 MULTIPLICATION BY A SCALARλ aaFigure 7.4Scalar multiplication of a vector (for λ>1).BµbpPλAOaFigure 7.5 An illustration of the ratio theorem. The point P divides the linesegment AB in the ratio λ : µ.Having defined the operations of addition, subtraction and multiplication by ascalar, we can now use vectors to solve simple problems in geometry.◮A point P divides a line segment AB in the ratio λ : µ (see figure 7.5). If the positionvectors of the points A and B are a and b, respectively, find the position vector of thepoint P .As is conventional for vector geometry problems, we denote the vector from the point Ato the point B by AB. If the position vectors of the points A and B, relative to some originO, area and b, it should be clear that AB = b − a.Now, from figure 7.5 we see that one possible way of reaching the point P from O isfirst to go from O to A and to go along the line AB for a distance equal to the the fractionλ/(λ + µ) of its total length. We may express this in terms of vectors asOP = p = a +λλ + µ AB= a + λ (b − a)=λ + µ(1 − λλ + µ= µλ + µ a +)a +λλ + µ bλ b, (7.6)λ + µwhich expresses the position vector of the point P in terms of those of A and B. Wewould,of course, obtain the same result by considering the path from O to B and then to P . ◭215