The Chemistry Maths Book, Second Edition

16.4 Scalar differentiation of a vector 453

(i)d 1 = 12 a 1 + 13 b 1 = 1 2(2i 1 + 13 j 1 + 1 k) 1 + 1 3(i 1 − 12 j) 1 = 1 (4i 1 + 16 j 1 + 12 k) 1 + 1 (3i 1 − 16 j)

= 17 i 1 + 12 k

(ii)Vectorb 1 = 1 i 1 − 12 jlies in thexy-plane, vectork lies along the z-direction. Therefore

λk(any λ) is perpendicular to b.

(iii) Vector dlies in the xz-plane, andλjis perpendicular to d.

0 Exercises 13, 14

Although the cartesian unit vectors provide the most widely used representation of

vectors in three dimensions, other representations are sometimes more useful. Thus,

any three noncoplanar vectors can be used as base. If a, band care three such vectors,

not necessarily unit vectors or perpendicular, then any other vector in the space can

be written

u 1 = 1 u

a

a 1 + 1 u

b

b 1 + 1 u

c

c (16.20)

The numbersu

a

,u

b

, andu

c

are the components ofu

along the directions of the base vectors. An example

of the use of non-cartesian bases is found in

crystallography in the description of the properties

of regular lattices. In this case a, band cdefine the

crystal axes and the unit cell, and every lattice point

has position vector (16.20) with components that

are integers. This is illustrated in Figure 16.15 for a

planar lattice.

16.4 Scalar differentiation of a vector

A vectora 1 = 1 a(t)is a function of the scalar variable

tif its magnitude or its direction, or both, depends

on the value of t. In Figure 16.16, and

are the vectors for values tandt 1 + 1 ∆t

of the variable. The change in the vector in interval

∆tis

∆a 1 = 1 a(t 1 + 1 ∆t) 1 − 1 a(t)

and the derivative of the vector with respect to tis

defined in the usual way by the limit (if the limit exists)

(16.21)

Taking the limit corresponds to letting point B approach point A along the curve

defined bya(t). The direction of approaches that of the tangent to the curve

at A, and this is the direction of the derivative at A.

AB =∆a

d

dt t

tt t

tt

aaaa

=

∆













=

+∆ −

∆

→→

lim lim

()()

∆∆ 00 tt





















OB =+a()tt∆

OA =a()t

•••

a

b

2 a+ 2 b

a+3b

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Figure 16.15

o

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a(t+∆ t)

∆ a

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Figure 16.16