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

7.2 Vectors and scalars: basic concepts 231
A
a 1 –2 b
2a
Figure7.16
Scalar multiplication ofavector.
b
C
b
M
Figure7.17
Thetriangle ABC,with midpointM
ofside AB.
a
B
→
Example7.5 In a triangle ABC, M is the midpoint of AB. Let AB be denoted by a, and BC → by b.
ExpressAC,
→ →
CA andCMinterms →
ofaandb.
Solution The situationissketched inFigure 7.17. Usingthe triangle ruleforaddition wefind
→
AB + → BC = → AC
Therefore,
→
AC=a+b
Itfollows that → CA=− → AC = −(a +b).
Again by the triangle ruleapplied totriangle CMB we find
→
CM = CB → + BM
→
→
NowBM →
= 1 BA=− 1 aand so
2 2
(
→
CM=−b+ − 1 )
2 a
=−
(b+ 1 )
2 a
7.2.6 Unitvectors
Vectors which have length 1 are called unit vectors. If a has length 3, for example,
then a unit vector in the direction of a is clearly 1 a. More generally we denote the unit
3
vectorinthedirectionabyâ.Recallthatthelengthormodulusofais |a|andsowecan
write
â = a
|a|
Note that |a|and hence 1
|a| arescalars.