>>11991233

The scalar product of two vectors measures the projection of one vector onto the other one. Since projection depends on the angle in Euclidean space, you can think about measuring the angle between the vectors. If they are 'similar' (= have similar direction), the projection will be large. If they are perpendicular to each other, projection is zero.

The concept is very useful because 1) it allows to talk about angles (or projections, rather) in vector spaces other than Euclidean; 2) any scalar (or, rather, inner) product induces norm which induces metric - and so we can talk about distances between abstract vectors.

One of the most useful examples is a vector space where vectors are functions (like cos(x)). As soon as you introduce an inner product, you can project one function onto another, and compute distances between functions. This allows us to perform analysis - convergence, Fourier series, functional derivatives, etc.

A vector product, by definition, is a vector in a direction that is orthogonal to both of the original vectors, and has length equal to the area of the parallelogram spanned by the two vectors. Vector product is an analog of a much more general concept of an exterior product which is something that allows us to compute volumes. Tthe exterior product on a generic vector space can only be defined for forms and not vectors (form is something that eats vectors and gives you a number, and is linear).

The result of an outer product is also a form and not a vector. In particular, a product of two forms is a 2-form which eats two vectors and gives oriented area of a parallelogram. This explains why vector product gives you area. To transform the product output (the 2-form) into a vector, you have to have some additional structure (like a scalar product). However, this only allows for assigning a vector to a 2-form in either a 3-dimensional or 7-dimensional space. Historically vector product appeared from Hamilton's quaternions.