The Chemistry Maths Book, Second Edition

16.6 The vector (cross) product 463

In terms of cartesian base vectors

The unit vectors i, jand kform a right-handed system of vectors and have properties

i 1 z 1 j 1 = 1 kj 1 z 1 k 1 = 1 ik 1 z 1 i 1 = 1 j

(16.51)

i 1 z 1 i 1 = 10 j 1 z 1 j 1 = 10 k 1 z 1 k 1 = 10

The vector producta 1 z 1 bcan then be expressed in terms of cartesian components:

a 1 z 1 b 1 = 1 (a

x

i 1 + 1 a

y

j 1 + 1 a

z

k) 1 z 1 (b

x

i 1 + 1 b

y

j 1 + 1 b

z

k)

= 1 a

x

b

x

i 1 z 1 i 1 + 1 a

x

b

y

i 1 z 1 j 1 + 1 a

x

b

z

i 1 z 1 k 1 + 1 a

y

b

x

j 1 z 1 i 1 + 1 a

y

b

y

j 1 z 1 j

• 1 a

y

b

z

j 1 z 1 k 1 + 1 a

z

b

x

k 1 z 1 i 1 + 1 a

z

b

y

k 1 z 1 j 1 + 1 a

z

b

z

k 1 z 1 k

= 1 a

x

b

y

k 1 + 1 a

x

b

z

(−j) 1 + 1 a

y

b

x

(−k) 1 + 1 a

y

b

z

i 1 + 1 a

z

b

x

j 1 + 1 a

z

b

y

(−i)

(remembering that, for example,j 1 z 1 i 1 = 1 −i 1 z 1 j). Therefore

a 1 z 1 b 1 = 1 (a

y

b

z

1 − 1 a

z

b

y

)i 1 + 1 (a

z

b

x

1 − 1 a

x

b

z

)j 1 + 1 (a

x

b

y

1 − 1 a

y

b

x

)k (16.52)

This form of the vector product can be written, and more easily remembered, in the

form of a determinant (Chapter 17),

(16.53)

EXAMPLE 16.15Givena 1 = 1 (3, 1, −1)andb 1 = 1 (1, 2, −3), finda 1 z 1 b,b 1 z 1 a, and the

area|a 1 z 1 b|of the parallelogram defined by aand b.

By equation (16.52),

= 1 −i 1 + 18 j 1 + 15 k 1 = 1 (−1, 8, 5)

Also,b 1 z 1 a 1 = 1 −a 1 z 1 b 1 = 1 (1, −8, −5)

0 Exercises 31–40

Vector products are used for the description of surfaces in geometry, and in the

vector integral calculus for the evaluation of surface integrals. In mechanics, the

vector product is used for the description of properties associated with torque and

angular motion.

||ab 11 z =++=185 90

222

ab11 11z =×−−−×i













+−×−×−







13 12 1133()() () ()







+×−×













jk 3211

ab

ijk

11 11z =aaa

bbb

x yz

xyz