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

y
j
O
i
r
Figure7.19
Thex--y plane with
point P.
P(x, y)
M
x
7.3 Cartesian components 233
to express r in terms of the numbers ofxandy. If we denote a unit vector along thex
axis by i, and a unit vector along theyaxis by j (we usually omit the ˆ here), then it is
clearfromthedefinitionofscalarmultiplicationthatOM =xi,andMP → =yj.Itfollows
fromthe triangle law of addition that
r = → OP = →
OM + → MP=xi+yj
Clearly the vectors i and j are orthogonal. The numbersxandyare the i and j componentsof
r. Furthermore, using Pythagoras’s theorem we can deduce that
r = √ x 2 +y 2
Alternative notations which aresometimes usefulare
(
r = OP → x
=
y)
→
and
r = → OP = (x,y)
( x
When written intheseforms iscalled a columnvector and (x,y) iscalled arow
y)
vector.Toavoidconfusionwiththecoordinates (x,y)weshallnotuserowvectorshere
but they will be needed in Chapter 26. We will also use the column vector notation for
moregeneralvectors, thus,
( a
ai+bj=
b)
Wesaidearlierthatavectorcanbetranslated,maintainingitslengthanddirectionwithoutchanging
the vector itself.While this is true generally, position vectors form animportant
exception. Position vectors are constrained to their specific position and must
always remaintiedtothe origin.
Example7.6 IfAisthepointwithcoordinates(5,4)andBisthepointwithcoordinates (−3,2)find
the position vectors ofAand B, and the vector → AB. Further, find | → AB|.
( 5
Solution The position vector of A is 5i +4j = , which we shall denote by a. The position
( ) 4)
−3
vector of B is −3i + 2j = , which we shall denote by b. Application of the
2
trianglelaw totriangleOAB (Figure 7.20)gives
thatis
→
OA + → AB = → OB
a + → AB=b