Final answer:
The cross product of two vectors results in a vector, is anticommutative, and distributive over addition, but it is not commutative or associative.
Step-by-step explanation:
In the context of vector multiplication, the cross product of two vectors results in a vector that is orthogonal to the plane formed by the original vectors. Unlike scalar multiplication, the cross product is not commutative; rather, it is anticommutative, meaning switching the order of the vectors negates the resulting vector. Also, the cross product is distributive, allowing the operation to be applied across a sum of vectors. When multiplied, vectors obey specific rules, such as the distributive property similar to the dot product. However, the cross product does not follow associative law, which pertains to vector addition and scalar vector multiplication.
The cross product AB is given by a determinant of unit vectors and scalar components, which is not associative. It’s crucial to maintain the multiplication order due to its anticommutative nature when performing cross product operations. For instance, given vectors A and B, the cross product AxB would equal −BxA. The result's direction is governed by the right-hand rule, resembling a corkscrew’s progression.