Answer
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The cross product, also known as the vector product, is an operation between two vectors in three-dimensional space that results in a third vector perpendicular to both input vectors. It is denoted by the symbol "×".
Given two vectors, A and B, the cross product is defined as:
A × B = |A| |B| sin(θ) n
where |A| and |B| are the magnitudes of vectors A and B, θ is the angle between the two vectors, and n is a unit vector perpendicular to the plane formed by A and B, following the right-hand rule.
The cross product satisfies the following algebraic laws:
1. Anticommutativity: A × B = - (B × A)
This means that the order of the vectors matters, and reversing the order changes the sign of the resulting vector.
2. Distributivity over vector addition: A × (B + C) = (A × B) + (A × C)
This law allows the cross product to distribute over vector addition.
3. Scalar multiplication: (kA) × B = A × (kB) = k(A × B)
The cross product can be multiplied by a scalar, and it can be factored out of a scalar multiplication.
However, the cross product does not satisfy the commutative law, meaning that A × B is not equal to B × A. This is evident from the anticommutativity law mentioned earlier.
The cross product of two vectors is equal to zero when the vectors are parallel or antiparallel to each other. In other words, if the angle between the vectors is either 0 degrees or 180 degrees, the cross product will be zero. For example, if A = (1, 2, 3) and B = (2, 4, 6), the angle between them is 0 degrees, and their cross product A × B will be equal to zero.
