Page 70 - Math SL HB Sem 2

Kolej MARA Seremban Mathematics IB High Level
Core : Vectors

The Scalar Product (Or Dot Product )

: :
The scalar product oftwo vectors is a ali + aj + a3k and b bi+ bi + b:k is the
operation which is written a . b and defined
a.b: lal lblcos0 ,0 is angle between a and D.

Proof

A

a b-a let OA =a and OB =b

... AB: AO + OB
ob B : -a+b
= b-a

By the cosine rule,

(Alf =toil,+toll,-z (OA)(OB) cos e
lu-al,: I"l,+ lalr- zl al lulcose
(b-a).(b-a) : a.a+b.b-21 al lblcoso

-
b . b b . a - a . b + a . a : a, a*b.t-2lal lnlcose
*2a.b: *zl al lblcose
.'. a.b: lal lulcose

The Angle Between Two Vectors

:
lf a= aj +a2i + a3k and b bi + bzi + b:k are two vectors and 0 is the angle between a
and b. From the definition of a . b ,
a.b:lallblcose
-
a.b
cosU=._
l,llrl
'
[ ,.r, )
... rhe angle 6 = cos-i
ti;JF|l

c orooerties of scalar prod for anv vector b and c and rn s scalar.

1) a.a= lal2

2) a.b= b.a
3) a.(b+c) =a.b*a.c