Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

7.6 MULTIPLICATION OF VECTORS


a

b

O


θ

bcosθ

Figure 7.8 The projection ofbonto the direction ofaisbcosθ. The scalar
product ofaandbisabcosθ.

in the previous example, the speed of the second particle relative to the first is


given by


u=|u|=


(−3)^2 +(−8)^2 =


73.

A vector whose magnitude equals unity is called aunit vector.The unit vector

in the directionais usually notatedaˆand may be evaluated as


aˆ=

a
|a|

. (7.14)


The unit vector is a useful concept because a vector written asλaˆthen has mag-


nitudeλand directionˆa. Thus magnitude and direction are explicitly separated.


7.6 Multiplication of vectors

We have already considered multiplying a vector by a scalar. Now we consider


the concept of multiplying one vector by another vector. It is not immediately


obvious what the product of two vectors represents and in fact two products


are commonly defined, thescalar productand thevector product. As their names


imply, the scalar product of two vectors is just a number, whereas the vector


product is itself a vector. Although neither the scalar nor the vector product


is what we might normally think of as a product, their use is widespread and


numerous examples will be described elsewhere in this book.


7.6.1 Scalar product

The scalar product (or dot product) of two vectorsaandbis denoted bya·b


and is given by


a·b≡|a||b|cosθ, 0 ≤θ≤π, (7.15)

whereθis the angle between the two vectors, placed ‘tail to tail’ or ‘head to head’.


Thus, the value of the scalar producta·bequals the magnitude ofamultiplied


by the projection ofbontoa(see figure 7.8).

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