2.3 Lengths of Vectors and Distances

Norms can be to compute the length of a vector. Inner products and norms are closely related in the sense that any inner product induces a norm

$\left \|  x \right \| := \sqrt{<x,x>}$

This shows that we can compute length of vectors using inner product.However not every norm is induced by an inner product. The Manhattan norm is an example of a norm without a corresponding inner product.Norms induced by inner products needs special attention

Cauchy-Schwarz Inequality

For an inner product vector space $( V, <\cdot,\cdot>)$, the induced norm $\left \|  \cdot \right \|$ satisfies the Cauchy-Schwarz Inequality  $|<x,y>|  \le \left \|  x \right \| \left \|  y \right \|$

Example :Length of vectors using inner product

In geometry, we are often interested in lengths of vectors. We can now use an inner product to compute the length. Let us take $x = [1,1]^T \in  \mathbb{R}^2$. If we use the dot product as the inner product, we obtain

$\left \|  x \right \| = \sqrt{x^Tx}=\sqrt{1^2+1^2}=\sqrt{2}$

Distance and Metric

Consider an inner product space $(V,<\cdot,\cdot>)$. Then

$d(x,y):=\left \|  x-y \right \| = \sqrt{<x-y,x-y>}$

is called the distance between $x$ and $y$ for $x,y \in V$.

If we use the dot product as the inner product, then the distance is called Euclidean distance.

The mapping

$d: V \times V \to \mathbb{R}$

$(x,y) \to d(x,y)$

is called a metric

The metric satisfies following properties

d is Positive definite i.e.,: $d(x,y) \ge 0$ for all $x,y \in V$  and $d(x,y)= 0  \Leftrightarrow x=y$

d is Symmetric i.e., $d(x,y)=d(y,x)$ for all $x,y \in V$

Triangle inequality:$d(x,z) <= d(x,y) + d(y,z)$ for all $x,y \in V$

Example:

if $x=[1,2]^T$ and $y=[2,3]^T$ in $\mathbb{R}^2$. Then $x-y=[-1,-1]^T$

$d(x,y)=\sqrt{(-1)^2+(-1)^2}=\sqrt{2}$

Compute the distance between $x=\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix},y=\begin{bmatrix} -1 \\ - 1\\ 0 \end{bmatrix}$ ( university question)

$x-y=(2,3,3)$    So the distance

$d(x,y)=\sqrt{2^2+3^2+3^2}=\sqrt{22}$

# calculating euclidean distance between vectors

from math import sqrt

import numpy as np

# define data

x =np.array([1,2])

y =np.array([2,3])

# calculate distance

d=x-y

print("x")

print(x)

print("y")

print(y)

print("x-y")

print(d)

dist = sqrt(d.dot(d))

print("distance")

print(dist)

O/P

x

[1 2] 

y 

[2 3] 

x-y 

[-1 -1] 

distance 

1.4142135623730951