Final answer:
The expressions (a × b) × (c × d) and a × (b × c) × d differ because cross product operations on vectors are neither associative nor commutative, leading to different results when the grouping or order of multiplication is altered.
Step-by-step explanation:
The expressions (a × b) × (c × d) and a × (b × c) × d are not the same due to the associative property of multiplication and the rules for cross products of vectors. The cross product is not associative; thus, the order in which the cross product is performed affects the result. Using the rule where (x^a)^b = x^{a.b}, we can see that taking the cross product of two vectors and then cross multiplying the result with another vector is not the same as changing the grouping of the vectors being cross multiplied.
For example, if you were to calculate the cross product of two vectors, A and B, to get a vector C (C = A × B), and then multiply C by another vector D, you would have (A × B) × D. However, if you take the cross product of B and C first and then cross multiply it with A, as in A × (B × C), you would generally end up with a different vector. This distinction arises because the cross product is not commutative nor associative; the sequence and grouping of multiplication matter.
It is critical to apply the associativity and commutative properties correctly when performing algebraic operations involving vectors to avoid incorrect results. In vector products, maintaining the proper multiplication order is essential because it can completely change the outcome.