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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

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