Sometimes on a single vector space, different norms can be defined. In this post, first we give the definition of an inner product vector space. Then, we proceed to discuss different norms on the finite-dimensional real vector spaces, i.e. the vector spaces of the form \(\mathbb{R}^n\). The dot product In the post on the dot…

# Tag: dot product

## Finite-dimensional vectors

In this post, we discuss algebra of finite-dimensional vectors and some fundamental concepts in linear algebra. First, we define finite-dimensional vectors. Let \(n\) be a positive integer. An \(n\)-dimensional real vector \(u\) is an ordered \(n\)-tuple (i.e., an ordered sequence with \(n\) elements) of the following form: $$ u = (u_1, \dots, u_n),$$ where \(u_i\)…

## Dot product of vectors

Definition of the dot product in the two-dimensional space The definition of the dot product of two vectors \(u = (u_1,u_2)\) and \(v = (v_1,v_2)\), denoted by \( u \cdot v \), is as follows: $$ u \cdot v = u_1 v_1 + u_2 v_2. $$ Example. Let us compute the dot product of \(u…