Use a specific example to prove that the cross product is also not associative. That is, use three specific vectors in 3-sapce to show that Vector a×(Vector b × Vector c) is not equal to (Vector a × Vector b) × Vector c.

To prove that the cross product is not associative, we need to provide a specific example where the two expressions yield different results.

Let's consider three specific vectors in 3-space:
Vector a = <1, 2, 3>
Vector b = <4, 5, 6>
Vector c = <7, 8, 9>

First, let's calculate Vector b × Vector c:
Vector b × Vector c = <4, 5, 6> × <7, 8, 9>

To find the cross product between b and c, we can use the determinant method:

i j k
4 5 6
7 8 9

= (5 * 9 - 6 * 8) i - (4 * 9 - 6 * 7) j + (4 * 8 - 5 * 7) k
= -3i + 6j - 3k

Therefore, Vector b × Vector c = <-3, 6, -3>.

Now, let's calculate Vector a × (Vector b × Vector c):
Vector a × (Vector b × Vector c) = <1, 2, 3> × <-3, 6, -3>

Again, we'll use the determinant method:

i j k
1 2 3
-3 6 -3

= (2 * -3 - 3 * 6) i - (1 * -3 - 3 * -3) j + (1 * 6 - 2 * -3) k
= -12i + 6j + 12k

Therefore, Vector a × (Vector b × Vector c) = <-12, 6, 12>.

Lastly, let's calculate (Vector a × Vector b) × Vector c:
(Vector a × Vector b) × Vector c = <1, 2, 3> × <4, 5, 6> × <7, 8, 9>

We can find (Vector a × Vector b) first:
(Vector a × Vector b) = <1, 2, 3> × <4, 5, 6>

Using the determinant method again:

i j k
1 2 3
4 5 6

= (2 * 6 - 3 * 5) i - (1 * 6 - 3 * 4) j + (1 * 5 - 2 * 4) k
= -3i - 6j - 3k

Therefore, (Vector a × Vector b) = <-3, -6, -3>.

Now we can calculate (Vector a × Vector b) × Vector c:
(<-3, -6, -3>) × <7, 8, 9>

Once again, using the determinant method:

i j k
-3 -6 -3
7 8 9

= (-6 * 9 - -3 * 8) i - (-3 * 9 - -3 * 7) j + (-3 * 8 - -6 * 7) k
= -15i + 15j - 9k

Therefore, (Vector a × Vector b) × Vector c = <-15, 15, -9>.

Comparing the results:
Vector a × (Vector b × Vector c) = <-12, 6, 12>
(Vector a × Vector b) × Vector c = <-15, 15, -9>

As we can see, the two expressions are not equal. Hence, we have shown a specific example where Vector a×(Vector b × Vector c) is not equal to (Vector a × Vector b) × Vector c, proving that the cross product is not associative.