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

8.4 Using matrices in the translation and rotation of vectors 267
c*a
ans =
-2 -6
-4 -8
Multiplication ofascalar byamatrixis
commutative.Itdoes notmatter in which order the
calculation is performed. Each element in the matrix
ismultiplied bythe scalar value.
3 Thecalculations are valid but they do notproduce the
same result becausematrixmultiplication isnot
commutative.
c*d
ans =
23 31
34 46
>> d*c
ans =
19 43
22 50
Thisresult emphasizes the need forcare when
manipulating matrixequations.
4 Bothcalculations are valid but the resultsare not the
same.
e*f
ans =
f*e
ans =
9 19 29
12 26 40
15 33 51
22 49
28 64
5 de isnot valid asthere are 2 columns ind and 3 rows
ine. Itshould generate an error in atechnical
computing language. Itisworth observing the error
message andunderstanding why thishappensfor
future reference.
ed isvalid,
e*d
ans =
29 39
40 54
51 69
6 Itis valid. We takethisto mean the matrix
multiplication ofcbyitself:
c*c
ans =
7 15
10 22
Notice that thisisdifferent to taking the squareof
each element withinc. Ifwe wanted to dothat it
would be necessaryto use the ‘dot’ notation:
c.^2
ans =
1 9
4 16
Thecommandc.ˆ2means‘takeeach elementofcand
raiseitto the power of2.There is asetofoperators
within MATLAB ® which allow multiplication of
matrices onan element byelement basis.Care has to
be taken as(forexample)c ∗dandc. ∗dare both
valid operationswhich generate a3by 3 result but the
elements ofthe matrixare different. Inother words,it
could give an incorrect result but MATLAB ® would
not giveany error messages.
7 Itis notvalid. Itis only possibleto multiply amatrix
by itselfifthere are the same number ofcolumns as
rows.
8.4 USINGMATRICESINTHETRANSLATIONANDROTATION
OFVECTORS
There are many applications in engineering that require vectors to be manipulated into
newconfigurations.Onemechanismforachievingthisisbytheuseofmatrices.Wewill
introduce this topic through the example of robotics. However, it is important to note
thatthe ideas aremore generally applicable inengineering.