The Chemistry Maths Book, Second Edition

450 Chapter 16Vectors (iii) Scalar multiplication.The product caof the scalar cand the vector ais obtained by multiplying each c ...

16.3 Components of vectors 451 EXAMPLE 16.5The centre of mass of a system of Nmasses, m 1 with position vectorr 1 (‘at positionr ...

452 Chapter 16Vectors A system of electric dipoles with momentsμ 1 , μ 2 , =, μ N , has total dipole moment (16.17) The total di ...

16.4 Scalar differentiation of a vector 453 (i)d 1 = 12 a 1 + 13 b 1 = 1 2(2i 1 + 13 j 1 + 1 k) 1 + 1 3(i 1 − 12 j) 1 = 1 (4i 1 ...

454 Chapter 16Vectors In terms of the components of the vector, ifa 1 = 1 a x i 1 + 1 a y j 1 + 1 a z k, wherei,j,andk are (cons ...

16.4 Scalar differentiation of a vector 455 where v x , v y and v z are the components of velocity in the three cartesian direct ...

456 Chapter 16Vectors (iv) The velocity in the x-direction isv x 1 = 14 so that the body is moving in the positive x-direction w ...

16.5 The scalar (dot) product 457 Proof To show the equivalence of the definitions (16.30) and (16.31), we apply the cosine rule ...

458 Chapter 16Vectors For nonzero vectors aand b, the scalar product is zero when the vectors are perpendicular, a 1 · 1 b 1 = 1 ...

16.5 The scalar (dot) product 459 The following examples demonstrate two applications of scalar products in the physical science ...

460 Chapter 16Vectors The general case If the force is notconstant along the path r 1 to r 2 then the work has the form of a lin ...

16.5 The scalar (dot) product 461 EXAMPLE 16.14Charges in an electric field The force experienced by a charge qin the presence o ...

462 Chapter 16Vectors 16.6 The vector (cross) product Every pair of non-parallel vectors, aand b, defines a parallelogram, Figur ...

16.6 The vector (cross) product 463 In terms of cartesian base vectors The unit vectors i, jand kform a right-handed system of v ...

464 Chapter 16Vectors EXAMPLE 16.16Moment of force (torque) In mechanics the magnitude of the moment of force F, or torque T, ab ...

16.6 The vector (cross) product 465 More generally, we consider a particle (that may be part of a rotating body) rotating about ...

466 Chapter 16Vectors where I 1 = 1 mr 2 is the moment of inertia about the axis of rotation. The relation between land yis less ...

16.8 The gradient of a scalar field 467 defines a vector associated with each point r. An example of a vector field is the veloc ...

468 Chapter 16Vectors EXAMPLE 16.20Force and potential energy By a generalization to three dimensions of the discussion of conse ...

16.9 Divergence and curl of a vector field 469 (16.71) is the electrostatic field of charge q 1 , andφ 1 = 1 V 2 q 2 1 = 1 q 1 2 ...