9.2: Calculating Vector Length, Normalization, Distance and Dot

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\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

#  Load Useful Python Libraries
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
from urllib.request import urlretrieve
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

In this section we will cover some of the basic vector math we will use this semester.

Do This

Watch the following summary video about calculation of vector length, Normalizing vectors and the distance between points then answer the questions.

from IPython.display import YouTubeVideo
YouTubeVideo("S0BIhbV6reI",width=640,height=360, cc_load_policy=True)

Vector:

\[(a_1, a_2, \dots a_n) \nonumber \]

\[(b_1, b_2, \dots b_n) \nonumber \]

Length:

\[length = \sqrt{a_1^2 + a_2^2 + \dots + a_n^2} \nonumber \]

Normalization:

\[\frac{1}{length}(a_1, a_2, \dots a_n) \nonumber \]

Distance:

\[distance = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2 + \dots + (a_n - b_n)^2} \nonumber \]

Question

Calculate length of vector (4.5, 2.6, 3.3, 4.1)?

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.float(length,'695da96d4a240e54bd8c61e75ff5a3e2');

Question

What is a normalized form of the vector (4.5, 2.6, 3.3, 4.1)?

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.vector(norm,'12c94f16ba11222987ca20006790182d');

Question

What is the distance between (4.5, 2.6, 3.3, 4.1) and (4, 3, 2, 1)?

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.float(distance,'d73defc9a514eb70434190e1757f5bb8');

Dot Product:

\[dot(a,b) = a_1b_1 + a_2b_2 +\dots + a_nb_n \nonumber \]

Do This

Review Sections 1.4 and 1.5 of the Boyd and Vandenberghe text and answer the questions below.

Question

What is the dot product between \(u=[1,7,9,11]\) and \(v=[7,1,2,2]\) (Store the information in a variable called uv)?

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.float(uv,'48044bf058c2d7d21b311b173a0ca7e5');

Question

What is the norm of vector \(u\) defined above (store this value in a variabled called n)?

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.float(n,'96078eb552924d7bdb9e67f9ecab88c1');

Question

What is the distance between points \(u\) and \(v\) defined above. (put your answer in a variable named d)

#Put your answer to the above question here

from answercheck import checkanswer

checkanswer.float(d,'71f49beeb28061bc60eb3d9966497416');