The Chemistry Maths Book, Second Edition

450 Chapter 16Vectors

(iii) Scalar multiplication.The product caof the scalar cand the vector ais obtained

by multiplying each component of aby c,

ca 1 = 1 (ca

x

, ca

y

, ca

z

) (16.11)

EXAMPLE 16.3

(i) Find the vector whose initial point P isp 1 = 1 (2, 1, 0)and whose terminal

point Q isq 1 = 1 (1, 3, −2). (ii) What is the length of a? (iii) Find the unit vector parallel

to a.

(i) a 1 = 1 q 1 − 1 p 1 = 1 (1, 3, −2) 1 − 1 (2, 1, 0) 1 = 1 (−1, 2, −2)

(ii)

(iii)â 1 = 1 a 2 |a| 1 = 1 (− 12 3, 2 2 3, − 22 3)

0 Exercises 2–5

EXAMPLE 16.4Given a 1 = 1 (2, 3, 1), b 1 = 1 (1, −2, 0), and c 1 = 1 (5, 2, −1), find (a)

d 1 = 12 a 1 + 13 b 1 − 1 cand (b)|d|.

(a) Ifd 1 = 1 (d

x

, d

y

,d

z

)then

d

x

1 = 12 a

x

1 + 13 b

x

1 − 1 c

x

1 = 1 (2 1 × 121 + 131 × 111 − 1 5) 1 = 12

d

y

1 = 12 a

y

1 + 13 b

y

1 − 1 c

y

1 = 1 (2 1 × 131 + 131 × 1 (−2) 1 − 1 2) 1 = 1 − 2

d

z

1 = 12 a

z

1 + 13 b

z

1 − 1 c

z

1 = 1 (2 1 × 111 + 131 × 101 − 1 (−1)) 1 = 13

andd 1 = 1 (2, −2, 3).

(b)

0 Exercises 6–10

Equation (16.9) for the equality of vectors and Example 16.4 show that, for vectors

in three dimensions, a vector equation is equivalent to three simultaneous scalar

equations, one for each component. Conversely, the three scalar equations are expressed

by the single vector equation. Vector algebra then provides a powerful method for

the formulation and solution of physical problems that involve vector quantities. The

resolution into the component equations is often necessary only when the solution

of a problem needs to be completed by the insertion of numerical values for the

components.

||d=++=+−+=ddd

xyz

222 2 22

22317()

|| ( )a=− ++− =12 2 3( )

22 2

a=PQ