Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.8 USING VECTORS TO FIND DISTANCES O P d a p nˆ Figure 7.15 The minimum distancedfrom a point to a plane.
Find the distance fr ...

VECTOR ALGEBRA O Q P ˆn q a b p Figure 7.16 The minimum distance from one line to another. Ifpandqare the position vectors of an ...

7.9 RECIPROCAL VECTORS the line to the plane is zero unless b·nˆ=0, in which case the distance,d, will be d=|(a−r)·ˆn|, whereris ...

VECTOR ALGEBRA not coplanar. Moreover, ifa,bandcare mutually orthogonal unit vectors then a′=a,b′=bandc′=c, so that the two syst ...

7.10 EXERCISES 7.2 A unit cell of diamond is a cube of sideA, with carbon atoms at each corner, at the centre of each face and, ...

VECTOR ALGEBRA 7.12 The planeP 1 contains the pointsA,BandC, which have position vectors a=− 3 i+2j,b=7i+2jandc=2i+3j+2k, respec ...

7.10 EXERCISES d a a b c Figure 7.17 A face-centred cubic crystal. points with coordinates X 1 =(3, 2 ,2),X 2 =(2, 3 ,1),X 3 =(2 ...

VECTOR ALGEBRA 7.22 In subsection 7.6.2 we showed how the moment or torque of a force about an axis could be represented by a ve ...

7.10 EXERCISES V 0 cosωt V 1 V 2 V 3 V 4 R 1 =50Ω R 2 I 1 I 2 I 3 L C=10μF Figure 7.18 An oscillatory electric circuit. The powe ...

VECTOR ALGEBRA of vector plots for potential differences and currents (they could all be on the same plot if suitable scales wer ...

8 Matrices and vector spaces In the previous chapter we defined avectoras a geometrical object which has both a magnitude and a ...

MATRICES AND VECTOR SPACES a discussion of how to use these properties to solve systems of linear equations. The application of ...

8.1 VECTOR SPACES the trivial case in which all the coefficients are zero) then the vectors arelinearly independent, and no vect ...

MATRICES AND VECTOR SPACES We reiterate that the vectorx(a geometrical entity) is independent of the basis it is only the compo ...

8.1 VECTOR SPACES In the above basis we may express any two vectorsaandbas a= ∑N i=1 aiˆei and b= ∑N i=1 bieˆi. Furthermore,in s ...

MATRICES AND VECTOR SPACES 8.1.3 Some useful inequalities For a set of objects (vectors) forming a linear vector space in which〈 ...

8.2 LINEAR OPERATORS where the equality holds if the sum includes allNbasis vectors. If not all the basis vectors are included i ...

MATRICES AND VECTOR SPACES may be thought of as ‘transforming’ one geometrical entity (i.e. a vector) into another. If we now in ...

8.3 MATRICES 8.2.1 Properties of linear operators Ifxis a vector andAandBare two linear operators then it follows that (A+B)x=Ax ...

MATRICES AND VECTOR SPACES In a similar way we may denote a vectorxin terms of its componentsxiin a basisei,i=1, 2 ,...,N, by th ...