What do symmetric bilinear forms have to do with the law of cosines? Does it have anything to do with the angle formula between two vectors?
Yes, symmetric bilinear forms are actually intimately connected to the law of cosines and the angle formula between two vectors.

The law of cosines states that in a triangle with sides a, b, and c, and angles A, B, and C opposite their respective sides, we have:

c^2 = a^2 + b^2 - 2ab*cos(C)

This formula relates the lengths of the sides of a triangle to the cosine of the angle between them. Interestingly, this formula can be generalized to higher dimensions using the notion of a dot product or inner product between two vectors.

In particular, given two vectors u and v in n-dimensional Euclidean space R^n, their dot product (also called their inner product) is defined as:

u · v = u_1v_1 + u_2v_2 + ... + u_n*v_n

This dot product satisfies many of the same properties as the usual multiplication of real numbers, including distributivity, commutativity, and associativity. Moreover, if we define the norm or length of a vector as:

||u|| = sqrt(u · u)

then we can prove that the law of cosines holds in this higher-dimensional setting as well. Specifically, if we let theta be the angle between two vectors u and v, then we have:

||u - v||^2 = ||u||^2 + ||v||^2 - 2||u||*||v||*cos(theta)

which is a higher-dimensional version of the law of cosines. This formula can be used to compute angles between vectors in any number of dimensions, not just the usual 2D or 3D cases.

Now, what does this have to do with symmetric bilinear forms? Well, it turns out that a symmetric bilinear form on a vector space V is essentially a generalization of the dot product or inner product. Specifically, a symmetric bilinear form is a function B: V x V -> R that satisfies the following properties:

1. B(u, v) = B(v, u) for all u, v in V (symmetry)
2. B(u+v, w) = B(u, w) + B(v, w) for all u, v, w in V (linearity in the first argument)
3. B(kv, w) = kB(v, w) for all v, w in V and scalar k in R (linearity in the first argument)

The dot product is a special case of a symmetric bilinear form, where the vector space is R^n and the bilinear form is given by:

(u, v) -> u · v

In fact, any symmetric bilinear form on R^n can be written in terms of the dot product using a matrix, and vice versa. This connection between bilinear forms and matrices is a central topic in linear algebra.

So, to answer your question, the angle formula between two vectors is intimately related to the dot product (or inner product) between them, which is a special case of a symmetric bilinear form. In this sense, symmetric bilinear forms are intimately connected to the law of cosines, and can be used to compute angles and distances in higher-dimensional vector spaces.
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