Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR ALGEBRAkaja z ka y ja x iiFigure 7.7 A Cartesian basis set. The vector a is the sum of a x i, a y j and a z k.basis vectors i, j and k may themselves be represented by (1, 0, 0), (0, 1, 0) and(0, 0, 1) respectively.We can consider the addition and subtraction of vectors in terms of theircomponents. The sum of two vectors a and b is found by simply adding theircomponents, i.e.a + b = a x i + a y j + a z k + b x i + b y j + b z k=(a x + b x )i +(a y + b y )j +(a z + b z )k, (7.11)and their difference by subtracting them,a − b = a x i + a y j + a z k − (b x i + b y j + b z k)=(a x − b x )i +(a y − b y )j +(a z − b z )k. (7.12)◮Two particles have velocities v 1 = i +3j +6k and v 2 = i − 2k, respectively. Find thevelocity u of the second particle relative to the first.The required relative velocity is given byu = v 2 − v 1 =(1− 1)i +(0− 3)j +(−2 − 6)k= −3j − 8k. ◭7.5 Magnitude of a vectorThe magnitude of the vector a is denoted by |a| or a. In terms of its componentsin three-dimensional Cartesian coordinates, the magnitude of a is given by√a ≡|a| = a 2 x + a 2 y + a 2 z. (7.13)Hence, the magnitude of a vector is a measure of its length. Such an analogy isuseful for displacement vectors but magnitude is better described, for example, by‘strength’ for vectors such as force or by ‘speed’ for velocity vectors. For instance,218