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

7.7 Vectors ofndimensions 253
and
H = − 2πE ( )
0 2πz
cos
λωµ 0 λ − ωt i
where λ, ω,µ 0 andE 0 are constants. Findthe
Poyntingvector andconfirm thatthe directionof
energy flow isthezdirection.
Solutions
1 (a) 2i−8j+k (b) 2i+13j+8k
(c) 4i −k (d) 22i −18j +12k
8
1
√
27
(i−5j+k)
2 (a) 5i+7j+3k (b) −5i−7j−3k
3 −10i −5j +10k
4 (a) −i−3j+2k (b) i−j
5 √ 89, √ 26,48.01,0.9980
6 (a) q×p (b) 2q×p
7
1
√
6
(i −j−2k),0.775
9 −i−46j+17k
11 (b) √ 90 = 9.49
12 5
13
4
√
3
(−2i −4j +6k)
( )
14 2πE2 0
cos 2 2πz
λωµ 0 λ − ωt k,
which is avector in thezdirection.
TechnicalComputingExercises7.6
Vectorandscalarproductsarereadilycalculatedinmost
technicalcomputing languages.Forexample in
MATLAB ® wecouldcalculatethe scalar product of
a=i−2j+3kandb=2i−j−kbytyping:
a=[1 -2 3];
b=[2 -1 -1];
dot(a,b)
producingthe answer:
ans= 1
andthe vector product bythentyping:
cross(a,b)
producingthe answer:
ans =
5 7 3
1 ConfirmusingMATLAB ® ,orasimilarlanguage,that
i×j=k.
2 (a) Selecttwo different vectorsaand b and
calculate |a ×b| 2 + (a.b) 2 .
(b) Calculate |a| 2 |b| 2
(c) Whatare yourobservations about the results
from (a)and(b)? Verifyyourconclusions by
changing the vectorsaandbandrepeating
the calculations.
7.7 VECTORSOFnDIMENSIONS
The examples we have discussed have all concerned two- and three-dimensional vectors.
Our understanding has been helped by the fact that two-dimensional vectors can
be drawn in the plane of the paper and three-dimensional vectors can be visualized in
thethree-dimensionalspaceinwhichwelive.However,therearesomesituationswhen