Question

Prove that the vector product (also called the cross product) of vectors in 3 dimensional space, does not satisfy an associative law for multiplication. (Hint: find a simple example of a cross product of 3 vectors that fails to be associative, showing the calculations.)

Answer 1

Answer:

a = (1,0,0)

b = (1,1,0)

c = (1,1,1)

Step-by-step explanation:

In order to do this, we must find a counter example, that is to say, three vectors a, b,and c such that

(a x b) x c different to a x (b x c)

Let a, b and c be the vectors

a = (1,0,0)

b = (1,1,0)

c = (1,1,1)

Then (a x b) x c = (-1,-1,0)

See attachment 1

whereas

a x (b x c) = (0,0,1)

See attachment 2

So,

$(a\times b)\times c \neq a \times(b\times c)$

and the cross product is not associative.