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

230 Chapter 7 Vectors
r 1
P
P
Q
P
PQ
Q
O
r 2
b
Q
p
O
q
r
R
O
p
q
Figure7.13
Thetwo vectors r 1 and r 2 .
Figure7.14
Thetetrahedron OPQR.
Figure7.15
Thetriangle OPQ.
Vectors do not necessarily lie in a two-dimensional plane. Three-dimensional vectors
are commonly used as isillustratedinthe following example.
Example7.4 OPQR is the tetrahedron shown in Figure 7.14. Let
→
ExpressPQ,
→ QRandRP → intermsofp, qandr.
Solution Consider the triangle OPQ shown in Figure 7.15. We note that
→
OP=p, → OQ=qand → OR=r.
→
OQ represents the third
sideofthetriangleformedwhenpand → PQareplacedheadtotail.Usingthetrianglelaw
we find
→
OP + → PQ = → OQ
Therefore,
→
PQ = → OQ − → OP
=q−p
→
Similarly, QR=r−qandRP=p−r.
→
7.2.5 Multiplicationofavectorbyascalar
Ifkis any positive scalar and a is a vector thenka is a vector in the same direction as a
butktimesaslong.Ifkisanynegative scalar,kaisavectorintheoppositedirectionto
a, andktimesas long.Bywayof example, studythe vectors inFigure 7.16. Clearly 2a
is twice as long as a but has the same direction. The vector 1 b is half as long as b but
2
has the same direction asb. Itispossible toprove the following rules.
For any scalarskandl, and any vectorsaand b:
k(a+b)=ka+kb
(k+l)a=ka+la
k(la) = (kl)a
Thevectorkaissaid tobeascalarmultiple ofa.