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

The vector product, or cross product, of vectors in 3-dimensional space does not satisfy an associative law for multiplication. This can be shown by calculating two different orders of the cross product and demonstrating that they are not equal.

Step-by-step explanation:

The vector product, or cross product, of vectors in 3-dimensional space does not satisfy an associative law for multiplication.

To prove this, let's take three vectors A, B, and C.

We will calculate two different orders of the cross product (A × B) × C and A × (B × C) and show that they are not equal.

Let's assume A = (1, 0, 0), B = (0, 1, 0), and C = (0, 0, 1).

(A × B) × C = ((AyBz - AzBy), (AzBx - AxBz), (AxBy - AyBx)) × C = (0, 0, -(1 * 1 - 0 * 0)) = (0, 0, -1)

A × (B × C) = A × ((BzCy - ByCz), (CxAz - CxAz), (AxBy - AyBx)) = (0,0, -(1* 0 - 0* 1)) = (0, 0, 0)

As we can see, the two results are different, hence the cross product does not satisfy an associative law for multiplication.

Answer 2

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* b)* c \\eq a *(b* c)`

and the cross product is not associative.