082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

254 Chapter 7 Vectors
the generalization to higher dimensions is appropriate, but no convenient geometrical
interpretation is available. Nevertheless, many of the concepts we have discussed are
still applicable. For example, wecan introduce the four-dimensional vectors
⎛ ⎞
⎛ ⎞
3 1
a = ⎜1
⎟
⎝2
⎠ and b= ⎜0
⎟
⎝3
⎠
4 1
It is natural to define the magnitude, or norm, of a as √ 3 2 +1 2 +2 2 +4 2 = √ 30 and
the scalar product of a and b as a·b = (3)(1) + (1)(0) + (2)(3) + (4)(1) = 13. An
n-dimensional vector will havencomponents. Operations such as addition, subtraction
and scalar multiplication aredefined inanobvious way.
It is also possible to define a set of variables as a vector. This turns out to be a useful
way of modelling a physical system. The system is described by means of a vector
which consists of an ordered set of variables sufficient to describe the state of the system.
Such a vector is called a state vector. This concept is explored in more detail in
Chapter 20.
Engineeringapplication7.12
Meshcurrentvector
When analysing a complex circuit it can be convenient to assign a current to each
smallindependentloopwithinthecircuit.Eachofthesecurrentsisknownasamesh
current. It is possible to collect these individual currents together to form a vector
quantity. Consider the following example.
A circuit as shown in Figure 7.33 has a set of mesh currents {I 1
,I 2
,I 3
,I 4
} from
whichwecan formacurrentvector
⎛ ⎞
I 1
I = ⎜I 2 ⎟
⎝I 3
⎠
I 4
Nogeometricalinterpretationispossiblebutneverthelessthisvectorprovidesauseful
mathematical way of handling the mesh currents. We shall see how vectors such
asthesecan bemanipulatedinSection8.12.
I 1 I 2 I 3 I 4
Figure7.33
Acircuitwith meshcurrents
shown.