Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.6 HINTS AND ANSWERS (a) LetRbe a real positive number and defineKmby Km= ∫R −R ( R^2 −x^2 )m dx. Show, using integration by pa ...

7 Vector algebra This chapter introduces space vectors and their manipulation. Firstly we deal with the description and algebra ...

7.2 ADDITION AND SUBTRACTION OF VECTORS a a b b a+b b+a Figure 7.1 Addition of two vectors showing the commutation relation. We ...

VECTOR ALGEBRA a a a b b b c c c a+(b+c) (a+b)+c b+c b+c a+b a+b Figure 7.2 Addition of three vectors showing the associativity ...

7.3 MULTIPLICATION BY A SCALAR a a λ Figure 7.4 Scalar multiplication of a vector (forλ>1). O A B P a b p μ λ Figure 7.5 An i ...

VECTOR ALGEBRA O A B C D E G F a b c Figure 7.6 The centroid of a triangle. The triangle is defined by the pointsA, BandCthat ha ...

7.4 BASIS VECTORS AND COMPONENTS These equations are consistent and have the solutionλ=μ=2/3. Substituting these values into eit ...

VECTOR ALGEBRA i j k axi ayj azk a Figure 7.7 A Cartesian basis set. The vectorais the sum ofaxi,ayjandazk. basis vectorsi,jandk ...

7.6 MULTIPLICATION OF VECTORS a b O θ bcosθ Figure 7.8 The projection ofbonto the direction ofaisbcosθ. The scalar product ofaan ...

VECTOR ALGEBRA From (7.15) we see that the scalar product has the particularly useful property that a·b= 0 (7.16) is a necessary ...

7.6 MULTIPLICATION OF VECTORS If we introduce a set of basis vectors that are mutually orthogonal, such asi,j, k, we can write t ...

VECTOR ALGEBRA θ a×b a b Figure 7.9 The vector product. The vectorsa,banda×bform a right-handed set. modate this extension the c ...

7.6 MULTIPLICATION OF VECTORS θ O R P F r Figure 7.10 The moment of the forceFaboutOisr×F. The cross represents the direction of ...

VECTOR ALGEBRA is the forward direction of a right-handed screw rotating in the same sense as the body. The velocity of any poin ...

7.6 MULTIPLICATION OF VECTORS O θ φ P a b c v Figure 7.11 The scalar triple product gives the volume of a parallelepiped. The sc ...

VECTOR ALGEBRA
Find the volumeVof the parallelepiped with sidesa=i+2j+3k,b=4i+5j+6kand c=7i+8j+10k. We have already found thata ...

7.7 EQUATIONS OF LINES, PLANES AND SPHERES O A R a b r Figure 7.12 The equation of a line. The vectorbis in the directionARand λ ...

VECTOR ALGEBRA O a d ˆn r AR Figure 7.13 The equation of the plane is (r−a)·ˆn=0. where the unit normal to the plane isˆn=li+mj+ ...

7.8 USING VECTORS TO FIND DISTANCES of the sphere. This is easily expressed in vector notation as |r−c|^2 =(r−c)·(r−c)=a^2 , (7. ...

VECTOR ALGEBRA O A P θ p−a d a b p Figure 7.14 The minimum distance from a point to a line.
Find the minimum distance from the ...