Vector Cross Product (AS)

In the context of Further Pure Mathematics 1, the vector cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space.
The cross product of two vectors results in a vector that is perpendicular (or orthogonal) to both of the original vectors, following the right-hand rule.
The right-hand rule states that if you point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, then your thumb will point in the direction of the cross product.
Unlike the dot product, the cross product is not commutative. This means that the order in which the operation is applied matters (i.e., a x b does not equal b x a). Instead, a x b is equal to -b x a.
The magnitude (or length) of the resulting vector from a cross product is equal to the area of the parallelogram that the two vectors would form if placed tail-to-tail.
The cross product can be calculated using a determinant of a 3x3 matrix, including the unit vectors (i,j,k) in the first row and the corresponding coordinates of the two vectors in the second and third rows, respectively.
The cross product of two parallel vectors is the zero vector because the parallelogram formed by two parallel vectors has no area.
The distributive law applies to the vector cross product, meaning that for three vectors a, b, and c: a x (b + c) equals a x b + a x c.
However, the scalar multiplication property also applies: for any scalar c and vectors a and b, the equation (ca) x b = a x (cb) = c(a x b) holds.
Understanding and applying the cross product, including the relevant properties and operations, contribute significantly towards mastering vector concepts in three-dimensional space, a key component of the Further Pure Mathematics 1 syllabus.