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

238 Chapter 7 Vectors
so thatadepends linearly oncandb. Providedk 2 ≠ 0,then
b = 1 k 2
c − k 1
k 2
a
so that b depends linearly on c and a. The set of vectors {a,b,c} is said to be linearly
dependent and any one of the set can be written as a linear combination of the other
two.Ingeneral, we have the following definition:
Asetofnvectors {a 1
,a 2
,...,a n
} islinearlydependentif the expression
k 1
a 1
+k 2
a 2
+···+k n
a n
=0
canbesatisfiedbyfindingscalarconstantsk 1
,k 2
,...,k n
,notallofwhicharezero.
If the only way we can make the combination zero is by choosing all thek i
s to be
zero,thenthe given setofvectors issaidtobe linearly independent.
Example7.10 Show thatthe vectors iandjarelinearly independent.
Solution Weformtheexpressionk 1
i +k 2
j = 0andtrytochoosek 1
andk 2
sothattheequationis
satisfied. Using columnvectors wehave
( ) ( ) ( 1 0 0
k 1
+k
0 2
=
1 0)
that is (k10 ) ( ) ( ) ( 0 k1 0
+ = =
k 2
k 2
0)
The only way we can satisfy the equation is by choosingk 1
= 0 andk 2
= 0 and hence
we conclude that the vectors i and j are linearly independent. Geometrically, we note
that sincethey are perpendicular, no scalar multiple ofican give j and viceversa.
Example7.11 Thevectors
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 5 13
⎝2⎠
⎝1⎠
⎝−1⎠
3 9 21
are linearlydependentbecause, forexample
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
5 1 13 0
3⎝1⎠ −2⎝2⎠ −1⎝−1⎠ = ⎝0⎠
9 3 21 0
EXERCISES7.3
1 PandQliein thex--y plane. Find → PQ,wherePis the
point with coordinates (5,1) andQisthe point with
coordinates (−1,4).Find | → PQ|.
2 AandBlie in thex--yplane. If Ais the point (3,4)
and Bis the point (1,−5) write down the vectors
→
OA, → OB and → AB.Find aunitvector in the direction
of → AB.