VEC-0040: Linear Combinations of Vectors

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\inputGif }[1]{The gif ‘‘#1" would be inserted here when publishing online. } \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\tdplotsinandcos }[3]{\pgfmathsetmacro {#1}{sin(#3)}\pgfmathsetmacro {#2}{cos(#3)}} \newcommand {\tdplotmult }[3]{\pgfmathsetmacro {#1}{#2*#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 {\tdplottransformmainrot }[3]{\tdplotcalctransformmainrot \par \pgfmathsetmacro {\tdplotresx }{\raaeul * #1 + \rabeul * #2 + \raceul * #3} \pgfmathsetmacro {\tdplotresy }{\rbaeul * #1 + \rbbeul * #2 + \rbceul * #3} \pgfmathsetmacro {\tdplotresz }{\rcaeul * #1 + \rcbeul * #2 + \rcceul * #3} } \newcommand {\tdplottransformrotmain }[3]{\tdplotcalctransformrotmain \par \pgfmathsetmacro {\tdplotresx }{\raaeul * #1 + \rabeul * #2 + \raceul * #3} \pgfmathsetmacro {\tdplotresy }{\rbaeul * #1 + \rbbeul * #2 + \rbceul * #3} \pgfmathsetmacro {\tdplotresz }{\rcaeul * #1 + \rcbeul * #2 + \rcceul * #3} } \newcommand {\tdplottransformmainscreen }[3]{\tdplotcalctransformmainscreen \par \pgfmathsetmacro {\tdplotresx }{\raarot * #1 + \rabrot * #2 + \racrot * #3} \pgfmathsetmacro {\tdplotresy }{\rbarot * #1 + \rbbrot * #2 + \rbcrot * #3} } \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 {\tdplotsetcoord }[4]{\tdplotsinandcos {\sinthetavec }{\costhetavec }{#3}\tdplotsinandcos {\sinphivec }{\cosphivec }{#4}\tdplotmult {\stcpv }{\sinthetavec }{\cosphivec }\tdplotmult {\stspv }{\sinthetavec }{\sinphivec }\coordinate (#1) at ($#2*(\stcpv ,\stspv ,\costhetavec )$); \coordinate (#1xy) at ($#2*(\stcpv ,\stspv ,0)$); \coordinate (#1xz) at ($#2*(\stcpv ,0,\costhetavec )$); \coordinate (#1yz) at ($#2*(0,\stspv ,\costhetavec )$); \coordinate (#1x) at ($#2*(\stcpv ,0,0)$); \coordinate (#1y) at ($#2*(0,\stspv ,0)$); \coordinate (#1z) at ($#2*(0,0,\costhetavec )$); } \newcommand {\tdplotsimplesetcoord }[4]{\tdplotsinandcos {\sinthetavec }{\costhetavec }{#3}\tdplotsinandcos {\sinphivec }{\cosphivec }{#4}\tdplotmult {\stcpv }{\sinthetavec }{\cosphivec }\tdplotmult {\stspv }{\sinthetavec }{\sinphivec }\coordinate (#1) at ($#2*(\stcpv ,\stspv ,\costhetavec )$); } \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 {\tdplotdosurfaceplot }[6]{\par \pgfmathsetmacro {\nextphi }{\curphi + \tdplotsuperfudge *\viewphistep } \par \begin {scope}[opacity=1] \par \par \tdplotcheckdiff {\nextphi }{360}{\origviewphistep }{#2}{} \tdplotcheckdiff {\nextphi }{0}{\origviewphistep }{#2}{} \par \tdplotcheckdiff {\nextphi }{90}{\origviewphistep }{#3}{} \tdplotcheckdiff {\nextphi }{450}{\origviewphistep }{#3}{} \end {scope} \par \foreach \curtheta in{\viewthetastart ,\viewthetainc ,...,\viewthetaend } { \par \pgfmathsetmacro {\curlongitude }{90 -\curphi } \pgfmathsetmacro {\curlatitude }{90 -\curtheta } \par \ifthenelse {\equal {\leftright }{-1.0}}{\pgfmathsetmacro {\curphi }{\curphi -\origviewphistep } }{} \par \pgfmathsetmacro {\tdplottheta }{mod(\curtheta ,360)} \pgfmathsetmacro {\tdplotphi }{mod(\curphi ,360)} \par \pgfmathparse {\tdplotphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplottheta >\tdplotuppertheta } \pgfmathsetmacro {\logictest }{1 -\pgfmathresult } \par \pgfmathparse {\tdplottheta <\tdplotlowertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathsetmacro {\tdplottheta }{\tdplottheta + \viewthetastep } \pgfmathparse {\tdplottheta >\tdplotuppertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplottheta <\tdplotlowertheta } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathsetmacro {\tdplotphi }{\tdplotphi + \viewphistep } \par \pgfmathparse {\tdplotphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \par \pgfmathsetmacro {\tdplottheta }{\curtheta } \pgfmathsetmacro {\tdplotphi }{\curphi } \par \ifthenelse {\equal {#6}{parametricfill}}{\ifthenelse {\equal {\logictest }{1.0}}{\pgfmathsetmacro {\radius }{#1} \pgfmathsetmacro {\tdplotr }{\radius *360} \par \pgfmathlessthan {\radius }{0} \pgfmathsetmacro {\phaseshift }{180 * \pgfmathresult } \par \pgfmathsetmacro {\colorarg }{#5} \pgfmathsetmacro {\colorarg }{\colorarg + \phaseshift } \pgfmathsetmacro {\colorarg }{mod(\colorarg ,360)} \par \pgfmathlessthan {\colorarg }{0} \pgfmathsetmacro {\colorarg }{\colorarg + 360*\pgfmathresult } \par \pgfmathdivide {\colorarg }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} }{}}{\pgfsetfillcolor {#5} } \pgfsetstrokecolor {#4} \par \ifthenelse {\equal {\leftright }{-1.0}}{\pgfmathsetmacro {\curphi }{\curphi + \origviewphistep } }{} \par \ifthenelse {\equal {\logictest }{1.0}}{\pgfmathsetmacro {\radius }{abs(#1)} \pgfpathmoveto {\pgfpointspherical {\curlongitude }{\curlatitude }{\radius }} \par \pgfmathsetmacro {\tdplotphi }{\curphi + \viewphistep } \pgfmathsetmacro {\radius }{abs(#1)} \pgfpathlineto {\pgfpointspherical {\curlongitude -\viewphistep }{\curlatitude }{\radius }} \par \pgfmathsetmacro {\tdplottheta }{\curtheta + \viewthetastep } \pgfmathsetmacro {\radius }{abs(#1)} \pgfpathlineto {\pgfpointspherical {\curlongitude -\viewphistep }{\curlatitude -\viewthetastep }{\radius }} \par \pgfmathsetmacro {\tdplotphi }{\curphi } \pgfmathsetmacro {\radius }{abs(#1)} \pgfpathlineto {\pgfpointspherical {\curlongitude }{\curlatitude -\viewthetastep }{\radius }} \pgfpathclose \par \pgfusepath {fill,stroke} }{} } } \newcommand {\tdplotshowargcolorguide }[4]{ \par \pgfmathsetmacro {\tdplotx }{#1} \pgfmathsetmacro {\tdploty }{#2} \pgfmathsetmacro {\tdplothuestep }{5} \pgfmathsetmacro {\tdplotxsize }{#3} \pgfmathsetmacro {\tdplotysize }{#4} \par \pgfmathsetmacro {\tdplotyscale }{\tdplotysize /360} \par \foreach \tdplotphi in {0,\tdplothuestep ,...,360} { \pgfmathdivide {\tdplotphi }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} \par \pgfmathsetmacro {\tdplotstarty }{\tdploty + \tdplotphi * \tdplotyscale } \pgfmathsetmacro {\tdplotstopy }{\tdplotstarty + \tdplothuestep * \tdplotyscale } \pgfmathsetmacro {\tdplotstartx }{\tdplotx } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \filldraw [tdplot_screen_coords] (\tdplotstartx ,\tdplotstarty ) rectangle (\tdplotstopx ,\tdplotstopy ); } \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )*\tdplotyscale } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \par \draw [tdplot_screen_coords] (\tdplotx ,\tdploty ) rectangle (\tdplotstopx ,\tdplotstopy ); \par \node [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdploty ) {$0$}; \node [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdplotstopy ) {$2\pi $}; \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )/2*\tdplotyscale } \node [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdplotstopy ) {$\pi $}; } \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 }[1]{\mathbf {#1}} \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 {\mathtoolsset }[1]{\setkeys {\MT_options_name: }{#1}} \newcommand {\refeq }[1]{\textup {\ref {#1}}} \newcommand {\lparen }[0]{(} \newcommand {\rparen }[0]{)} \newcommand {\ordinarycolon }[0]{:} \newcommand {\MT_test_for_tcb_other:nnnnn }[1]{\MH_if:w t#1\relax \expandafter \MH_use_choice_i:nnnn \MH_else: \MH_if:w c#1\relax \expandafter \expandafter \expandafter \MH_use_choice_ii:nnnn \MH_else: \MH_if:w b#1\relax \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \MH_use_choice_iii:nnnn \MH_else: \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \MH_use_choice_iv:nnnn \MH_fi: \MH_fi: \MH_fi: } \newcommand {\newcases }[6]{\newenvironment {#1}{\MT_start_cases:nnnn {#2}{#3}{#4}{#5}}{\MH_end_cases: \right #6}} \newcommand {\renewcases }[6]{\renewenvironment {#1}{\MT_start_cases:nnnn {#2}{#3}{#4}{#5}}{\MH_end_cases: \right #6}} \newcommand {\SwapAboveDisplaySkip }[0]{\noalign {\vskip -\abovedisplayskip \vskip \abovedisplayshortskip }} \newcommand {\vdotswithin }[1]{{\mathmakebox [\widthof {\ensuremath {{}#1{}}}][c]{{\vdots }}}} \newcommand {\MTFlushSpaceBelow }[0]{\\\noalign {\nobreak \vskip -\lineskip \vskip -\l_MT_shortvdotswithinadjustbelow_dim \vskip -\origjot \vskip \jot }} \newcommand {\mathmbox }[0]{\mathpalette \MT_mathmbox:nn } \newcommand {\prescript }[3]{\mathchoice {\MT_prescript_inner: {#1}{#2}{#3}{\scriptstyle }}{\MT_prescript_inner: {#1}{#2}{#3}{\scriptstyle }}{\MT_prescript_inner: {#1}{#2}{#3}{\scriptscriptstyle }}{\MT_prescript_inner: {#1}{#2}{#3}{\scriptscriptstyle }}} \newcommand {\spreadlines }[1]{\setlength {\jot }{#1}\ignorespaces } \newcommand {\newgathered }[4]{\newenvironment {#1}{\def \MT_gathered_pre: {#2}\def \MT_gathered_post: {#3}\def \MT_gathered_env_end: {#4}\MT_gathered_env }{\endMT_gathered_env }} \newcommand {\renewgathered }[4]{\renewenvironment {#1}{\def \MT_gathered_pre: {#2}\def \MT_gathered_post: {#3}\def \MT_gathered_env_end: {#4}\MT_gathered_env }{\endMT_gathered_env }} \newcommand {\lgathered }[0]{\def \MT_gathered_pre: {}\def \MT_gathered_post: {\hfil }\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\rgathered }[0]{\def \MT_gathered_pre: {\hfil }\def \MT_gathered_post: {}\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\gathered }[0]{\def \MT_gathered_pre: {\hfil }\def \MT_gathered_post: {\hfil }\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\splitfrac }[2]{\genfrac {}{}{0pt}{1}{\textstyle #1\quad \hfill }{\textstyle \hfill \quad \mathstrut #2}} \newcommand {\splitdfrac }[2]{\genfrac {}{}{0pt}{0}{#1\quad \hfill }{\hfill \quad \mathstrut #2}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\dblcolon }[0]{\vcentcolon \mathrel {\mkern -.9mu}\vcentcolon } \newcommand {\coloneqq }[0]{\vcentcolon \mathrel {\mkern -1.2mu}=} \newcommand {\Coloneqq }[0]{\dblcolon \mathrel {\mkern -1.2mu}=} \newcommand {\coloneq }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\mathrel {-}} \newcommand {\Coloneq }[0]{\dblcolon \mathrel {\mkern -1.2mu}\mathrel {-}} \newcommand {\eqqcolon }[0]{=\mathrel {\mkern -1.2mu}\vcentcolon } \newcommand {\Eqqcolon }[0]{=\mathrel {\mkern -1.2mu}\dblcolon } \newcommand {\eqcolon }[0]{\mathrel {-}\mathrel {\mkern -1.2mu}\vcentcolon } \newcommand {\Eqcolon }[0]{\mathrel {-}\mathrel {\mkern -1.2mu}\dblcolon } \newcommand {\colonapprox }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\approx } \newcommand {\Colonapprox }[0]{\dblcolon \mathrel {\mkern -1.2mu}\approx } \newcommand {\colonsim }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\sim } \newcommand {\Colonsim }[0]{\dblcolon \mathrel {\mkern -1.2mu}\sim } \newcommand {\nuparrow }[0]{\MH_nuparrow: } \newcommand {\ndownarrow }[0]{\MH_ndownarrow: } \newcommand {\bigtimes }[0]{\MH_csym_bigtimes: }$

We define a linear combination of vectors and examine whether a given vector may be expressed as a linear combination of other vectors, both algebraically and geometrically.

VEC-0040: Linear Combinations of Vectors

A vector $\vec {v}$ is said to be a linear combination of vectors $\vec {v}_1, \vec {v}_2,\ldots , \vec {v}_n$ if $\vec {v}=a_1\vec {v}_1+ a_2\vec {v}_2+\ldots + a_n\vec {v}_n$ for some scalars $a_1, a_2, \ldots ,a_n$ .

For example, $\begin {bmatrix} -4\\9\\-10\\-1\end {bmatrix}$ is a linear combination of $\begin {bmatrix} -1\\3\\-3\\0\end {bmatrix}$ , $\begin {bmatrix} 2\\0\\1\\4\end {bmatrix}$ and $\begin {bmatrix} 0\\1\\-1\\1\end {bmatrix}$ because $\begin {bmatrix} -4\\9\\-10\\-1\end {bmatrix}=2\begin {bmatrix} -1\\3\\-3\\0\end {bmatrix}+(-1)\begin {bmatrix} 2\\0\\1\\4\end {bmatrix}+3\begin {bmatrix} 0\\1\\-1\\1\end {bmatrix}$

In this module we will focus on vectors in $\RR ^2$ and $\RR ^3$ .

Geometry of Linear Combinations

Express $\begin {bmatrix}2\\4\end {bmatrix}$ as a linear combination of $\begin {bmatrix}2\\1\end {bmatrix}$ and $\begin {bmatrix}2\\-2\end {bmatrix}$ . Interpret your results geometrically.

We need to find scalars $a$ and $b$ such that $\begin {bmatrix}2\\4\end {bmatrix}=a\begin {bmatrix}2\\1\end {bmatrix}+b\begin {bmatrix}2\\-2\end {bmatrix}$ This amounts to solving a system of linear equations $\begin{align*} 2a+2b&=2\\ a-2b&=4 \end{align*}$

Use your favorite method to solve this system. (Hint: adding the second equation to the first will work well for this system.) You will find that $a=2$ and $b=-1$ . Now we can write $\begin {bmatrix}2\\4\end {bmatrix}$ as a linear combination of $\begin {bmatrix}2\\1\end {bmatrix}$ and $\begin {bmatrix}2\\-2\end {bmatrix}$ as follows:

$\begin {bmatrix}2\\4\end {bmatrix}=2\begin {bmatrix}2\\1\end {bmatrix}+(-1)\begin {bmatrix}2\\-2\end {bmatrix}$

Geometrically speaking, this means that the vector $\begin {bmatrix}2\\4\end {bmatrix}$ is the diagonal of the parallelogram determined by $2\begin {bmatrix}2\\1\end {bmatrix}$ and $(-1)\begin {bmatrix}2\\-2\end {bmatrix}$ . The original vectors $\begin {bmatrix}2\\1\end {bmatrix}$ and $\begin {bmatrix}2\\-2\end {bmatrix}$ are shown below together with the parallelogram and its diagonal.

Use geometry to express $\begin {bmatrix}3\\-4\end {bmatrix}$ as a linear combination of $\begin {bmatrix}-1\\3\end {bmatrix}$ and $\begin {bmatrix}-4\\2\end {bmatrix}$ .

In addition to the Explanation below, the following video illustrates this process.

We are looking for $\begin {bmatrix}3\\-4\end {bmatrix}$ to be the diagonal of a parallelogram determined by scalar multiples of $\begin {bmatrix}-1\\3\end {bmatrix}$ and $\begin {bmatrix}-4\\2\end {bmatrix}$ .

Because a scalar multiple of a vector can point in the same direction as the vector or in the opposite direction, we will start by drawing straight lines determined by the two vectors.

The two lines that we drew will contain the sides of the parallelogram we are looking for. To find the other two sides we will draw lines parallel to $\begin {bmatrix}-1\\3\end {bmatrix}$ and $\begin {bmatrix}-4\\2\end {bmatrix}$ through the head of vector $\begin {bmatrix}3\\-4\end {bmatrix}$ .

Now the parallelogram is clearly visible.

The last remaining task is to identify the sides of the parallelogram as scalar multiples of $\begin {bmatrix}-1\\3\end {bmatrix}$ and $\begin {bmatrix}-4\\2\end {bmatrix}$ . We do this by identifying vectors $\begin {bmatrix}1\\-3\end {bmatrix}$ and $\begin {bmatrix}2\\-1\end {bmatrix}$ as the vectors that determine the parallelogram.

Observe that vector $\begin {bmatrix}2\\-1\end {bmatrix}$ is half the length of $\begin {bmatrix}-4\\2\end {bmatrix}$ and points in the opposite direction, while the vector $\begin {bmatrix}1\\-3\end {bmatrix}$ is the same length as $\begin {bmatrix}-1\\3\end {bmatrix}$ and also points in the opposite direction.

Now we can write $\begin {bmatrix}3\\-4\end {bmatrix}$ as a linear combination of $\begin {bmatrix}-1\\3\end {bmatrix}$ and $\begin {bmatrix}-4\\2\end {bmatrix}$ as follows $\begin {bmatrix}3\\-4\end {bmatrix}=\begin {bmatrix}1\\-3\end {bmatrix}+\begin {bmatrix}2\\-1\end {bmatrix}=(-1)\begin {bmatrix}-1\\3\end {bmatrix}+\left (-\frac {1}{2}\right )\begin {bmatrix}-4\\2\end {bmatrix}$

The method we used in Example ex:lincombparallelogrammethod to express the given vector as a linear combination of two other vectors is sufficiently useful that we summarize the steps.

Given two non-collinear vectors $\vec {u}$ and $\vec {v}$ in $\RR ^2$ , and a vector $\vec {w}$ which is not a scalar multiple of either $\vec {u}$ or $\vec {v}$ , we can express $\vec {w}$ as a linear combination of $\vec {u}$ and $\vec {v}$ as follows:

(a): Draw lines $L_{\vec {u}}$ and $L_{\vec {v}}$ determined by $\vec {u}$ and $\vec {v}$ , respectively.
(b): Through the head of vector $\vec {w}$ , draw lines $P_{\vec {u}}$ and $P_{\vec {v}}$ , parallel to $L_{\vec {u}}$ and $L_{\vec {v}}$ , respectively.
(c): Let $A$ be the point of intersection of $P_{\vec {u}}$ and $L_{\vec {v}}$ .
(d): Let $B$ be the point of intersection of $P_{\vec {v}}$ and $L_{\vec {u}}$ .
(e): Let $O$ denote the origin. Then $\overrightarrow {OA}=k_1\vec {v}$ and $\overrightarrow {OB}=k_2\vec {u}$ for some scalars $k_1$ and $k_2$ .
(f): We have $\vec {w}=k_2\vec {u}+k_1\vec {v}$ .

Can the vector $\begin {bmatrix}3\\2\end {bmatrix}$ be written as a linear combination of vectors $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ ?

We will start with a geometric approach.

Observe that $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ are scalar multiples of each other and lie on the same line.

A linear combination of $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ has the form:

$a\begin {bmatrix}6\\-2\end {bmatrix}+b\begin {bmatrix}-3\\1\end {bmatrix}=a(-2)\begin {bmatrix}-3\\1\end {bmatrix}+b\begin {bmatrix}-3\\1\end {bmatrix}=(-2a+b)\begin {bmatrix}-3\\1\end {bmatrix}$

This shows that all linear combinations of $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ will be scalar multiples of $\begin {bmatrix}-3\\1\end {bmatrix}$ , and therefore lie on the same line as $\begin {bmatrix}-3\\1\end {bmatrix}$ . Since $\begin {bmatrix}3\\2\end {bmatrix}$ does not lie on the line determined by $\begin {bmatrix}-3\\1\end {bmatrix}$ it cannot be expressed as a linear combination of $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ .

We can also address this question algebraically. To express $\begin {bmatrix}3\\2\end {bmatrix}$ as a linear combination of $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ , we need to solve the equation. $a\begin {bmatrix}-3\\1\end {bmatrix}+b\begin {bmatrix}6\\-2\end {bmatrix}=\begin {bmatrix}3\\2\end {bmatrix}$ This gives us a system of equations

$\begin{align*} -3a+6b&=3\\ a-2b&=2 \end{align*}$

When you try to solve this system using your favorite method, you will find that the system is inconsistent. Thus, $\begin {bmatrix}3\\2\end {bmatrix}$ cannot be written as a linear combination of $\begin {bmatrix}-3\\1\end {bmatrix}$ and $\begin {bmatrix}6\\-2\end {bmatrix}$ .

Algebra of Linear Combinations

If possible, express $\begin {bmatrix}7\\4\\-5\end {bmatrix}$ as a linear combination of $\begin {bmatrix}1\\-2\\1\end {bmatrix}$ and $\begin {bmatrix}3\\0\\-1\end {bmatrix}$ .

We are looking for coefficients $a$ and $b$ such that $a\begin {bmatrix}1\\-2\\1\end {bmatrix}+b\begin {bmatrix}3\\0\\-1\end {bmatrix}=\begin {bmatrix}7\\4\\-5\end {bmatrix}$ This translates into a system of equations $\begin{align*} a+3b&=7\\ -2a&=4\\ a-b&=-5 \end{align*}$

Solve this system for $a$ and $b$ , and enter your answers below: $a=\answer {-2}\quad \text {and}\quad b=\answer {3}$ We conclude that $\begin {bmatrix}7\\4\\-5\end {bmatrix}$ is a linear combination of $\begin {bmatrix}1\\-2\\1\end {bmatrix}$ and $\begin {bmatrix}3\\0\\-1\end {bmatrix}$ , and write: $\begin {bmatrix}7\\4\\-5\end {bmatrix}=\answer {-2}\begin {bmatrix}1\\-2\\1\end {bmatrix}+\answer {3}\begin {bmatrix}3\\0\\-1\end {bmatrix}$

Set up a system of equations that can be used to express $\begin {bmatrix}2\\-1\\3\\0\end {bmatrix}$ as a linear combination of $\begin {bmatrix}1\\0\\4\\-2\end {bmatrix}$ , $\begin {bmatrix}-2\\-1\\1\\-1\end {bmatrix}$ , $\begin {bmatrix}0\\4\\-3\\1\end {bmatrix}$ and $\begin {bmatrix}1\\1\\-1\\4\end {bmatrix}$ , or to determine that such a combination does not exist. Do not solve the system.

We are looking for $x_1$ , $x_2$ , $x_3$ and $x_4$ such that $x_1\begin {bmatrix}1\\0\\4\\-2\end {bmatrix}+x_2\begin {bmatrix}-2\\-1\\1\\-1\end {bmatrix}+x_3\begin {bmatrix}0\\4\\-3\\1\end {bmatrix}+x_4\begin {bmatrix}1\\1\\-1\\4\end {bmatrix}=\begin {bmatrix}2\\-1\\3\\0\end {bmatrix}$ This translates into the following system of equations:

$\begin {array}{ccccccccc} x_1 & -&2x_2 & & &+ & x_4&= &2 \\ & &-x_2 &+ &4x_3 & +& x_4&= &-1 \\ 4x_1 &+ &x_2 & -&3x_3 &-&x_4&= &3\\ -2x_1& -&x_2 & +&x_3&+&4x_4&=&0 \end {array}$ At this point you may not have the techniques for solving this system of equations efficiently. So, we present this problem as an example of a set-up only. Later in the course you will learn to solve and interpret such systems.

Practice Problems

Solve a system of linear equations to express $\begin {bmatrix}-1\\7\end {bmatrix}$ as a linear combination of $\begin {bmatrix}1\\2\end {bmatrix}$ and $\begin {bmatrix}-1\\1\end {bmatrix}$ .

System of linear equations: $\begin {array}{ccccc} \answer {1}a & +&\answer {-1}b&= &\answer {-1} \\ \answer {2}a& +&\answer {1}b&=&\answer {7} \end {array}$

Values of $a$ and $b$ : $a=\answer {2}\quad \text {and}\quad b=\answer {3}$

Linear Combination: $\begin {bmatrix}-1\\7\end {bmatrix}=\answer {2}\begin {bmatrix}1\\2\end {bmatrix}+\answer {3}\begin {bmatrix}-1\\1\end {bmatrix}$

Use Procedure pro:lincombgeo to express $\begin {bmatrix}-3\\0\end {bmatrix}$ as a linear combination of $\begin {bmatrix}2\\4\end {bmatrix}$ and $\begin {bmatrix}-1\\1\end {bmatrix}$ .

Linear Combination: $\begin {bmatrix}-3\\0\end {bmatrix}=\answer {-0.5}\begin {bmatrix}2\\4\end {bmatrix}+\answer {2}\begin {bmatrix}-1\\1\end {bmatrix}$

Use two different approaches (algebraic and geometric) to explain why the vector $\begin {bmatrix}5\\1\end {bmatrix}$ cannot be expressed as a linear combination of vectors $\begin {bmatrix}2\\-1\end {bmatrix}$ and $\begin {bmatrix}-4\\2\end {bmatrix}$ .

Set up a system of equations that can be used to express $\begin {bmatrix}1\\2\\-3\end {bmatrix}$ as a linear combination of vectors $\begin {bmatrix}1\\1\\0\end {bmatrix}$ , $\begin {bmatrix}3\\-1\\1\end {bmatrix}$ and $\begin {bmatrix}-2\\-1\\0\end {bmatrix}$ .

System of Equations: $\begin {array}{ccccccc} \answer {1}x_1 & +&\answer {3}x_2&+&\answer {-2}x_3&= &\answer {1} \\ \answer {1}x_1& +&\answer {-1}x_2&+&\answer {-1}x_3&=&\answer {2}\\ \answer {0}x_1& +&\answer {1}x_2&+&\answer {0}x_3&=&\answer {-3} \end {array}$