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 a×( b × c ) is not equal to ( a ×b ) × c.
so, have you picked out three vectors?
To demonstrate that the cross product is not associative, we need to find three specific vectors in 3D space that do not follow the associative property. Let's consider the vectors u = [1, 0, 0], v = [0, 1, 0], and w = [0, 0, 1].
First, let's calculate a × (b × c):
a = u = [1, 0, 0]
b = v = [0, 1, 0]
c = w = [0, 0, 1]
Now, let's evaluate b × c:
b × c = [0, 1, 0] × [0, 0, 1] = [(1 × 0) - (0 × 0), (0 × 0) - (0 × 1), (0 × 1) - (1 × 0)] = [0, 0, 0]
Therefore, a × (b × c) = [1, 0, 0] × [0, 0, 0] = [0, 0, 0].
Now, let's calculate (a × b) × c:
a = u = [1, 0, 0]
b = v = [0, 1, 0]
c = w = [0, 0, 1]
Let's evaluate a × b:
a × b = [1, 0, 0] × [0, 1, 0] = [(0 × 0) - (1 × 1), (1 × 0) - (0 × 0), (0 × 1) - (1 × 0)] = [-1, 0, 0]
Now, (a × b) × c:
(a × b) × c = [-1, 0, 0] × [0, 0, 1] = [(0 × 0) - (0 × 1), (0 × 0) - (-1 × 1), (-1 × 1) - (0 × 0)] = [0, 1, -1]
Therefore, a × (b × c) = [0, 0, 0] and (a × b) × c = [0, 1, -1].
From the calculations, we can clearly see that a × (b × c) is not equal to (a × b) × c since [0, 0, 0] ≠ [0, 1, -1]. Hence, this example demonstrates that the cross product is not associative.