18670f…: “Outside a magenta theorem”

$\newenvironment {prompt}{}{} \newcommand {\accordion }[0]{} \newcommand {\ungraded }[0]{} \newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex} \newcommand {\xmDefaultPrintstyle }[0]{xmPrintstyle.sty} \newcommand {\tdplotsinandcos }[3]{\pgfmathsetmacro {#1}{sin(#3)}\pgfmathsetmacro {#2}{cos(#3)}} \newcommand {\tdplotdiv }[3]{\pgfmathsetmacro {#1}{#2/#3}} \newcommand {\tdplotcheckdiff }[5]{\par \par \pgfmathparse { abs(#2 - #1)<#3 } \par \ifthenelse {\equal {\pgfmathresult }{1}}{#4}{#5} } \newcommand {\tdplotsetmaincoords }[2]{\pgfmathsetmacro {\tdplotmaintheta }{#1} \pgfmathsetmacro {\tdplotmainphi }{#2} \tdplotcalctransformmainscreen \tikzset {tdplot_main_coords/.style={x={(\raarot cm,\rbarot cm)},y={(\rabrot cm, \rbbrot cm)},z={(\racrot cm, \rbcrot cm)}}}} \newcommand {\tdplotcalctransformmainscreen }[0]{\tdplotsinandcos {\sintheta }{\costheta }{\tdplotmaintheta }\tdplotsinandcos {\sinphi }{\cosphi }{\tdplotmainphi }\tdplotmult {\stsp }{\sintheta }{\sinphi }\tdplotmult {\stcp }{\sintheta }{\cosphi }\tdplotmult {\ctsp }{\costheta }{\sinphi }\tdplotmult {\ctcp }{\costheta }{\cosphi }\pgfmathsetmacro {\raarot }{\cosphi }\pgfmathsetmacro {\rabrot }{\sinphi }\pgfmathsetmacro {\racrot }{0}\pgfmathsetmacro {\rbarot }{-\ctsp }\pgfmathsetmacro {\rbbrot }{\ctcp }\pgfmathsetmacro {\rbcrot }{\sintheta }\pgfmathsetmacro {\rcarot }{\stsp }\pgfmathsetmacro {\rcbrot }{-\stcp }\pgfmathsetmacro {\rccrot }{\costheta }} \newcommand {\tdplotcalctransformrotmain }[0]{\tdplotsinandcos {\sinalpha }{\cosalpha }{\tdplotalpha } \tdplotsinandcos {\sinbeta }{\cosbeta }{\tdplotbeta } \tdplotsinandcos {\singamma }{\cosgamma }{\tdplotgamma } \tdplotmult {\sasb }{\sinalpha }{\sinbeta } \tdplotmult {\sbsg }{\sinbeta }{\singamma } \tdplotmult {\sasg }{\sinalpha }{\singamma } \tdplotmult {\sasbsg }{\sasb }{\singamma } \tdplotmult {\sacb }{\sinalpha }{\cosbeta } \tdplotmult {\sacg }{\sinalpha }{\cosgamma } \tdplotmult {\sbcg }{\sinbeta }{\cosgamma } \tdplotmult {\sacbsg }{\sacb }{\singamma } \tdplotmult {\sacbcg }{\sacb }{\cosgamma } \tdplotmult {\casb }{\cosalpha }{\sinbeta } \tdplotmult {\cacb }{\cosalpha }{\cosbeta } \tdplotmult {\cacg }{\cosalpha }{\cosgamma } \tdplotmult {\casg }{\cosalpha }{\singamma } \tdplotmult {\cacbsg }{\cacb }{\singamma } \tdplotmult {\cacbcg }{\cacb }{\cosgamma } \pgfmathsetmacro {\raaeul }{\cacbcg - \sasg } \pgfmathsetmacro {\rabeul }{-\cacbsg - \sacg } \pgfmathsetmacro {\raceul }{\casb } \pgfmathsetmacro {\rbaeul }{\sacbcg + \casg } \pgfmathsetmacro {\rbbeul }{-\sacbsg + \cacg } \pgfmathsetmacro {\rbceul }{\sasb } \pgfmathsetmacro {\rcaeul }{-\sbcg } \pgfmathsetmacro {\rcbeul }{\sbsg } \pgfmathsetmacro {\rcceul }{\cosbeta } } \newcommand {\tdplotcalctransformmainrot }[0]{\tdplotsinandcos {\sinalpha }{\cosalpha }{\tdplotalpha } \tdplotsinandcos {\sinbeta }{\cosbeta }{\tdplotbeta } \tdplotsinandcos {\singamma }{\cosgamma }{\tdplotgamma } \tdplotmult {\sasb }{\sinalpha }{\sinbeta } \tdplotmult {\sbsg }{\sinbeta }{\singamma } \tdplotmult {\sasg }{\sinalpha }{\singamma } \tdplotmult {\sasbsg }{\sasb }{\singamma } \tdplotmult {\sacb }{\sinalpha }{\cosbeta } \tdplotmult {\sacg }{\sinalpha }{\cosgamma } \tdplotmult {\sbcg }{\sinbeta }{\cosgamma } \tdplotmult {\sacbsg }{\sacb }{\singamma } \tdplotmult {\sacbcg }{\sacb }{\cosgamma } \tdplotmult {\casb }{\cosalpha }{\sinbeta } \tdplotmult {\cacb }{\cosalpha }{\cosbeta } \tdplotmult {\cacg }{\cosalpha }{\cosgamma } \tdplotmult {\casg }{\cosalpha }{\singamma } \tdplotmult {\cacbsg }{\cacb }{\singamma } \tdplotmult {\cacbcg }{\cacb }{\cosgamma } \pgfmathsetmacro {\raaeul }{\cacbcg - \sasg } \pgfmathsetmacro {\rabeul }{\sacbcg + \casg } \pgfmathsetmacro {\raceul }{-\sbcg } \pgfmathsetmacro {\rbaeul }{-\cacbsg - \sacg } \pgfmathsetmacro {\rbbeul }{-\sacbsg + \cacg } \pgfmathsetmacro {\rbceul }{\sbsg } \pgfmathsetmacro {\rcaeul }{\casb } \pgfmathsetmacro {\rcbeul }{\sasb } \pgfmathsetmacro {\rcceul }{\cosbeta } } \newcommand {\tdplotsetrotatedcoords }[3]{\pgfmathsetmacro {\tdplotalpha }{#1} \pgfmathsetmacro {\tdplotbeta }{#2} \pgfmathsetmacro {\tdplotgamma }{#3} \tdplotcalctransformrotmain \par \tdplotmult {\raaeaa }{\raarot }{\raaeul } \tdplotmult {\rabeba }{\rabrot }{\rbaeul } \tdplotmult {\raceca }{\racrot }{\rcaeul } \tdplotmult {\raaeab }{\raarot }{\rabeul } \tdplotmult {\rabebb }{\rabrot }{\rbbeul } \tdplotmult {\racecb }{\racrot }{\rcbeul } \tdplotmult {\raaeac }{\raarot }{\raceul } \tdplotmult {\rabebc }{\rabrot }{\rbceul } \tdplotmult {\racecc }{\racrot }{\rcceul } \tdplotmult {\rbaeaa }{\rbarot }{\raaeul } \tdplotmult {\rbbeba }{\rbbrot }{\rbaeul } \tdplotmult {\rbceca }{\rbcrot }{\rcaeul } \tdplotmult {\rbaeab }{\rbarot }{\rabeul } \tdplotmult {\rbbebb }{\rbbrot }{\rbbeul } \tdplotmult {\rbcecb }{\rbcrot }{\rcbeul } \tdplotmult {\rbaeac }{\rbarot }{\raceul } \tdplotmult {\rbbebc }{\rbbrot }{\rbceul } \tdplotmult {\rbcecc }{\rbcrot }{\rcceul } \pgfmathsetmacro {\raarc }{\raaeaa + \rabeba + \raceca } \pgfmathsetmacro {\rabrc }{\raaeab + \rabebb + \racecb } \pgfmathsetmacro {\racrc }{\raaeac + \rabebc + \racecc } \pgfmathsetmacro {\rbarc }{\rbaeaa + \rbbeba + \rbceca } \pgfmathsetmacro {\rbbrc }{\rbaeab + \rbbebb + \rbcecb } \pgfmathsetmacro {\rbcrc }{\rbaeac + \rbbebc + \rbcecc } \tikzset {tdplot_rotated_coords/.append style={x={(\raarc cm,\rbarc cm)},y={(\rabrc cm, \rbbrc cm)},z={(\racrc cm, \rbcrc cm)}}}} \newcommand {\tdplotsetrotatedcoordsorigin }[1]{\tikzset {tdplot_rotated_coords/.append style={shift=#1}}} \newcommand {\tdplotresetrotatedcoordsorigin }[0]{\tikzset {tdplot_rotated_coords/.append style={shift={(0,0,0)}}}} \newcommand {\tdplotsetthetaplanecoords }[1]{\tdplotresetrotatedcoordsorigin \tdplotsetrotatedcoords {270 + #1}{270}{0}} \newcommand {\tdplotsetrotatedthetaplanecoords }[1]{\tdplotsetrotatedcoords {\tdplotalpha }{\tdplotbeta }{\tdplotgamma + #1}\tikzset {tdplot_rotated_coords/.append style={y={(\raarc cm,\rbarc cm)},z={(\rabrc cm, \rbbrc cm)},x={(\racrc cm, \rbcrc cm)}}}} \newcommand {\tdplotsetpolarplotrange }[4]{\pgfmathsetmacro {\tdplotlowerphi }{#3} \pgfmathsetmacro {\tdplotupperphi }{#4} \pgfmathsetmacro {\tdplotlowertheta }{#1} \pgfmathsetmacro {\tdplotuppertheta }{#2} } \newcommand {\tdplotresetpolarplotrange }[0]{\pgfmathsetmacro {\tdplotlowerphi }{0} \pgfmathsetmacro {\tdplotupperphi }{360} \pgfmathsetmacro {\tdplotlowertheta }{0} \pgfmathsetmacro {\tdplotuppertheta }{180} } \newcommand {\tdplotgetpolarcoords }[3]{\pgfmathsetmacro {\vxcalc }{#1} \pgfmathsetmacro {\vycalc }{#2} \pgfmathsetmacro {\vzcalc }{#3} \pgfmathsetmacro {\vcalc }{ sqrt((\vxcalc )^2 + (\vycalc )^2 + (\vzcalc )^2) } \par \pgfmathsetmacro {\vxycalc }{ sqrt((\vxcalc )^2 + (\vycalc )^2) } \par \pgfmathsetmacro {\tdplotrestheta }{asin(\vxycalc /\vcalc )} \pgfmathparse {\vzcalc < 0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotrestheta }{180 - \tdplotrestheta } } {} \ifthenelse {\equal {\vxcalc }{0.0}}{\pgfmathparse {\vycalc < 0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{270} } {\pgfmathparse {\vycalc > 0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{90} } {\pgfmathsetmacro {\tdplotresphi }{0} } } } {\pgfmathsetmacro {\tdplotresphi }{atan(\vycalc /\vxcalc )} \pgfmathparse {\vxcalc < 0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{\tdplotresphi +180} } { } \par \pgfmathparse {\tdplotresphi < 0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{\tdplotresphi +360} } {} } } \newcommand {\vec }[0]{\mathbf } \newcommand {\RR }[0]{\mathbb {R}} \newcommand {\dfn }[0]{\textit } \newcommand {\dotp }[0]{\cdot } \newcommand {\id }[0]{\text {id}} \newcommand {\norm }[1]{\left \lVert #1\right \rVert } \newcommand {\luastring }[1]{"\luatexluaescapestring {#1}"} \newcommand {\luastringO }[1]{\luastring {\unexpanded \expandafter {#1}}} \newcommand {\luastringN }[1]{\luastring {\unexpanded {#1}}} \newcommand {\RestoreMathJaxEnvironment }[1]{\expandafter \let \csname #1\expandafter \endcsname \csname mathjax-#1\endcsname \expandafter \let \csname end#1\expandafter \endcsname \csname mathjax-end#1\endcsname } \newcommand {\DeclareMicrotypeVariants }[1]{} \newcommand {\DeclareMicrotypeAlias }[2]{} \newcommand {\LoadMicrotypeFile }[1]{} \newcommand {\DeclareMicrotypeFilePrefix }[1]{} \newcommand {\DeclareMicrotypeBabelHook }[2]{} \newcommand {\microtypesetup }[1]{} \newcommand {\microtypecontext }[1]{} \newcommand {\textmicrotypecontext }[2]{#2} \newcommand {\leftprotrusion }[1]{#1} \newcommand {\rightprotrusion }[1]{#1} \newcommand {\noprotrusionifhmode }[0]{} \newcommand {\lsstyle }[0]{} \newcommand {\lslig }[1]{#1} \newcommand {\maketitle }[0]{} \newcommand {\problem }[0]{ } \newcommand {\exercise }[0]{ } \newcommand {\question }[0]{ } \newcommand {\exploration }[0]{ } \newcommand {\hint }[0]{ } \newcommand {\ConfigureTheoremEnv }[1]{\renewenvironment {#1}[1][]{\refstepcounter {problem}\ifthenelse {\equal {###1}{}}{}{\HCode {<span class="theorem-like-title">}###1\HCode {</span>}}}{} \ConfigureEnv {#1}{\stepcounter {identification}\ifvmode \IgnorePar \fi \EndP \HCode {<div class="theorem-like problem-environment #1" id="problem\arabic {identification}" titletext=" \GetTranslation {#1}">}}{\ifvmode \IgnorePar \fi \EndP \HCode {</div>}\IgnoreIndent }{}{}} \newcommand {\dialogue }[0]{\begin {description}} \newcommand {\foldable }[0]{\stepcounter {identification}\ifvmode \IgnorePar \fi \EndP \HCode {<div id="foldable\arabic {identification}" class="foldable">}} \newcommand {\unfoldable }[1]{\HCode {<span class="unfoldable">}#1\HCode {</span>}} \newcommand {\selectAll }[0]{\refstepcounter {problem}} \newcommand {\freeResponse }[0]{\refstepcounter {problem}} \newcommand {\javascript }[0]{\NoFonts } \newcommand {\sageCell }[0]{\NoFonts } \newcommand {\sageOutput }[0]{\NoFonts } \newcommand {\sagesilent }[0]{\NoFonts } \newcommand {\ungraded }[0]{\ifvmode \IgnorePar \fi \EndP \HCode {<div class="ungraded">}\IgnoreIndent } \newcommand {\accordion }[0]{\ifvmode \IgnorePar \fi \EndP \HCode {<div class="accordion">} } \newcommand {\expandable }[2]{\stepcounter {identification}\ifvmode \IgnorePar \fi \EndP \HCode {<div class="accordion">\Hnewline }\HCode {<h3 class="#1">}#2\HCode {</h3>\Hnewline }\HCode {<div id="accordion-item\arabic {identification}" class="accordion-item #1-content">\Hnewline } } \newcommand {\paragraph }[1]{\HCode {<span class="paragraphHead">}#1\HCode {</span>}\par \IgnorePar } \newcommand {\subparagraph }[1]{\HCode {<span class="subparagraphHead">}#1\HCode {</span>}\par \IgnorePar } \newcommand {\outcome }[1]{\HCode {<span class="learning-outcome">#1</span>} } \newcommand {\javascriptCode }[0]{\NoFonts } \newcommand {\GetTranslation }[1]{#1} \newcommand {\ref }[1]{\HCode {<a class="reference" href="\##1">#1</a>}}$

Use the navigation bar arrows in the interactive below to advance the construction steps. Use the final result to express $\vec {w}$ as a linear combination of $\vec {u}$ and $\vec {v}$.

\[\vec {w}=\answer {-1}\vec {u}+\answer {0.5}\vec {v}\]