30. We consider all possible products and then simplify using relations such as ˆi

· ˆk = 0 and ˆi· ˆi = 1. Thus,

a

·
b =

a

x

ˆi+ a

y

ˆj + a

z

ˆ

k



·

b

x

ˆi+ b

y

ˆj + b

z

ˆ

k



=

a

x

b

x

ˆi

· ˆi+ a

x

b

y

ˆi

· ˆj+ a

x

b

z

ˆi

· ˆk + a

y

b

x

ˆj

· ˆi+ a

y

b

j

ˆj

· ˆj+ · · ·

=

a

x

b

x

(1) + a

x

b

y

(0) + a

x

b

z

(0) + a

y

b

x

(0) + a

y

b

j

(1) +

· · ·

which is seen to reduce to the desired result (one might wish to show this in two dimensions before
tackling the additional tedium of working with these three-component vectors).