Visualizing the Dot Product

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\mathrm tr}} \newcommand{\CC}{{\mathbb C}} \newcommand{\HH}{{\mathbb H}} \newcommand{\KK}{{\mathbb K}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf A} \newcommand{\BB}{\vf B} \newcommand{\CCv}{\vf C} \newcommand{\EE}{\vf E} \newcommand{\FF}{\vf F} \newcommand{\GG}{\vf G} \newcommand{\HHv}{\vf H} \newcommand{\II}{\vf I} \newcommand{\JJ}{\vf J} \newcommand{\KKv}{\vf Kv} \renewcommand{\SS}{\vf S} \renewcommand{\aa}{\VF a} \newcommand{\bb}{\VF b} \newcommand{\ee}{\VF e} \newcommand{\gv}{\VF g} \newcommand{\iv}{\vf\imath} \newcommand{\rr}{\VF r} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF u} \newcommand{\vv}{\VF v} \newcommand{\ww}{\VF w} \newcommand{\grad}{\vf\nabla} \newcommand{\zero}{\vf 0} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1/rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \renewcommand{\ii}{\xhat} \renewcommand{\jj}{\yhat} \renewcommand{\kk}{\zhat} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 3.8 Visualizing the Dot Product

The figure below shows two computations of the dot product of the given vectors, using the geometric formula (top) and the algebraic formula (bottom). The magnitude of each vector and the angle between them are shown for the first computation, whereas the second shows the horizontal and vertical components. The components are color-coded; the green bars in the algebraic representation show the products of corresponding components.

Figure 3.8.1. Two representations of the dot product. Change the vectors by dragging their tips.