1. Defining scalar products in vector spaces allows you to introduce notions of angle between two vectors, and also gives a new definition of the length of a vector.

Note that the definition of the vector space is primary, you need not have an inner product always. So there are vector spaces which do not carry an inner product. One example is $L^p$ spaces with $p \neq 2$. Also note that this is an example where the vector space is normed, but the norm (size) is not associated with an inner product.

There are close connections between some kinds of continuous groups (Lie Groups) and vector spaces. There is a Lie Algebra (a vector space) associated with every Lie Group.

Defining scalar products in vector spaces allows you to introduce notions of angle between two vectors, and also gives a definition of the length of a vector.

1. Defining scalar products in vector spaces allows you to introduce notions of length and angle, for instance $<v,v> = ||v||^2$ Note that the definition of the vector space is primary, you need not have an inner product always.

There are close connections between some kinds of continuous groups (Lie Groups) and vector spaces. There is a Lie Algebra (a vector space) associated with every Lie Group.

1. Defining scalar products in vector spaces allows you to introduce notions of angle between two vectors, and also gives a new definition of the length of a vector.

Note that the definition of the vector space is primary, you need not have an inner product always. So there are vector spaces which do not carry an inner product. One example is $L^p$ spaces with $p \neq 2$. Also note that this is an example where the vector space is normed, but the norm (size) is not associated with an inner product.

There are close connections between some kinds of continuous groups (Lie Groups) and vector spaces. There is a Lie Algebra (a vector space) associated with every Lie Group.