Understanding Engineering Mathematics

The notationa∧bis sometimes used.
This vector product is often represented in the form of the array shown below, in order
to aid memory.

a×b≡

∣

∣

∣

∣

∣

ijk

a 1 a 2 a 3

b 1 b 2 b 3

∣ ∣ ∣ ∣ ∣ ≡

∣

∣

∣

∣

a 2 a 3

b 2 b 3

∣

∣

∣

∣i−

∣

∣

∣

∣

a 1 a 3

b 1 b 3

∣

∣

∣

∣j+

∣

∣

∣

∣

a 1 a 2

b 1 b 2

∣

∣

∣

∣k

where we introduce the shorthand
∣
∣
∣∣ab
cd

∣

∣

∣∣=ad−bc

NB: This is just a mnemonic device for remembering the formula for the vector product
and for calculating it – you don’t need to know anything about determinants at this stage.

Problem 11.9

Find the vector product a×b between the vectors a=i−jY2k and b=

2iYk. What is b×a?

Using the component expression given we obtain (check these calculations against the
symbolic expressions given above)

a×b=

∣

∣

∣

∣∣

ijk

1 − 12

201

∣

∣

∣

∣∣

=[(− 1 )( 1 )− 2 ( 0 )]i−[1× 1 − 2 ×2]j+[1× 0 −(− 1 )( 2 )]k

=−i+ 3 j+ 2 k

You can also verify explicitly that

b×a=i− 3 j− 2 k=−a×b

From the above definition it can be shown that the vector (orcross) product has the
following properties:

(i) a×bis a vector

(ii) a×(b+c)=a×b+a×cand(a+b)×c=a×c+b×c

i.e. the vector product is distributive over vector addition

(iii) (ka)×b=k(a×b)=a×(kb)for any scalark

(iv) Ifa×b=0anda,bare not zero vectors, thenaandbare parallel

(v) b×a=−a×bas illustrated by Problem 11.9

(vi) a×a=0, which follows from (v)

(vii) The vector product is not associative, i.e.

a×(b×c)=(a×b)×c