The result of the outer product operation is a vector, the length of which is equal to the modular product of the original vector, and the direction is perpendicular to the plane of the original vector. If the lengths of the original vectors A and B are A and B respectively, then the length of the new vector C is ab.
An important property of the outer product operation is the exchange law, that is, a _ b = b _ a, which means that no matter how we exchange the order of two vectors, the result is the same.
Another important property of the outer product operation is the distribution law, that is, (a+b) _ c = a _ c+b _ c, which means that we can decompose a vector into several parts, then perform the outer product operation with another vector respectively, and finally add these results.
A common application of outer product operation is to calculate the normal vector of two vectors in three-dimensional space. The normal vector is a vector perpendicular to the plane of the original vector, and its direction can be obtained by the cross product of the original vector. For example, if we have two vectors A=(a, b, c) and B=(d, e, f), then their normal vectors N=A×B=(b*f-c*e, c*d-a*f, a*e-b*d).
Generally speaking, the formula of outer product operation is a very powerful tool, which can help us solve many complicated mathematical problems. By understanding and mastering this formula, we can better understand and deal with the problems in vector space.