$\require{mathtools} % %%% GENERIC MATH %%% % % Environments \newcommand{\al}[1]{\begin{align}#1\end{align}} % need this for \tag{} to work \renewcommand{\r}{\mathrm} % % Greek \newcommand{\eps}{\epsilon} \newcommand{\veps}{\varepsilon} \newcommand{\O}{\Omega} \newcommand{\o}{\omega} \let\fi\phi % because it looks like an f \let\phi\varphi % because it looks like a p % % Miscellaneous shortcuts % .. over and under \newcommand{\ss}[1]{_{\substack{#1}}} \newcommand{\ob}{\overbrace} \newcommand{\ub}{\underbrace} \newcommand{\ol}{\overline} \newcommand{\tld}{\widetilde} \newcommand{\HAT}{\widehat} \newcommand{\f}{\frac} \newcommand{\s}[2]{#1 /\mathopen{}#2} \newcommand{\rt}{\sqrt} % .. relations \newcommand{\sr}{\stackrel} \newcommand{\sse}{\subseteq} \newcommand{\ce}{\coloneqq} \newcommand{\ec}{\eqqcolon} \newcommand{\ap}{\approx} \newcommand{\ls}{\lesssim} \newcommand{\gs}{\greatersim} % .. miscer \newcommand{\q}{\quad} \newcommand{\qq}{\qquad} \newcommand{\heart}{\heartsuit} % % Delimiters % (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing) % .. one-part \newcommand{\p}[1]{\mathopen{}\left( #1 \right)} \newcommand{\b}[1]{\mathopen{}\left[ #1 \right]} \newcommand{\set}[1]{\mathopen{}\left\{ #1 \right\}} \newcommand{\abs}[1]{\mathopen{}\left\lvert #1 \right\rvert} \newcommand{\floor}[1]{\mathopen{}\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\mathopen{}\left\lceil #1 \right\rceil} \newcommand{\inner}[1]{\mathopen{}\left\langle #1 \right\rangle} % .... (use phantom to force at least the standard height of double bars) \newcommand{\norm}[1]{\mathopen{}\left\lVert #1 \vphantom{f} \right\rVert} \newcommand{\frob}[1]{\norm{#1}_\mathrm{F}} %% .. two-part \newcommand{\incond}[2]{#1 \mathop{}\middle|\mathop{} #2} \newcommand{\cond}[2]{ {\left.\incond{#1}{#2}\right.}} \newcommand{\pco}[2]{\p{\incond{#1}{#2}}} \newcommand{\bco}[2]{\b{\incond{#1}{#2}}} \newcommand{\setco}[2]{\set{\incond{#1}{#2}}} \newcommand{\at}[2]{\left.#1\right|_{#2}} \newcommand{\pat}[2]{\p{\at{#1}{#2}}} \newcommand{\bat}[2]{\b{\at{#1}{#2}}} % ..... (use phantom to force at least the standard height of double bar) \newcommand{\oldpara}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2} %\newcommand{\para}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2} \newcommand{\para}[2]{\mathchoice{\begin{matrix}#1\\\hdashline#2\end{matrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}} \newcommand{\ppa}[2]{\p{\para{#1}{#2}}} \newcommand{\bpa}[2]{\b{\para{#1}{#2}}} %\newcommand{\bpaco}[4]{\bpa{\incond{#1}{#2}}{\incond{#3}{#4}}} \newcommand{\bpaco}[4]{\bpa{\cond{#1}{#2}}{\cond{#3}{#4}}} % % Levels of closeness \newcommand{\scirc}[1]{\sr{\circ}{#1}} \newcommand{\sdot}[1]{\sr{.}{#1}} \newcommand{\slog}[1]{\sr{\log}{#1}} \newcommand{\createClosenessLevels}[7]{ \newcommand{#2}{\mathrel{(#1)}} \newcommand{#3}{\mathrel{#1}} \newcommand{#4}{\mathrel{#1\!\!#1}} \newcommand{#5}{\mathrel{#1\!\!#1\!\!#1}} \newcommand{#6}{\mathrel{(\sdot{#1})}} \newcommand{#7}{\mathrel{(\slog{#1})}} } \let\lt\undefined \let\gt\undefined % .. vanilla versions (is it within a constant?) \newcommand{\ez}{\scirc=} \newcommand{\eq}{\simeq} \newcommand{\eqq}{\mathrel{\eq\!\!\eq}} \newcommand{\eqqq}{\mathrel{\eq\!\!\eq\!\!\eq}} \newcommand{\lez}{\scirc\le} \newcommand{\lq}{\preceq} \newcommand{\lqq}{\mathrel{\lq\!\!\lq}} \newcommand{\lqqq}{\mathrel{\lq\!\!\lq\!\!\lq}} \newcommand{\gez}{\scirc\ge} \newcommand{\gq}{\succeq} \newcommand{\gqq}{\mathrel{\gq\!\!\gq}} \newcommand{\gqqq}{\mathrel{\gq\!\!\gq\!\!\gq}} \newcommand{\lz}{\scirc<} \newcommand{\lt}{\prec} \newcommand{\ltt}{\mathrel{\lt\!\!\lt}} \newcommand{\lttt}{\mathrel{\lt\!\!\lt\!\!\lt}} \newcommand{\gz}{\scirc>} \newcommand{\gt}{\succ} \newcommand{\gtt}{\mathrel{\gt\!\!\gt}} \newcommand{\gttt}{\mathrel{\gt\!\!\gt\!\!\gt}} % .. dotted versions (is it equal in the limit?) \newcommand{\ed}{\sdot=} \newcommand{\eqd}{\sdot\eq} \newcommand{\eqqd}{\sdot\eqq} \newcommand{\eqqqd}{\sdot\eqqq} \newcommand{\led}{\sdot\le} \newcommand{\lqd}{\sdot\lq} \newcommand{\lqqd}{\sdot\lqq} \newcommand{\lqqqd}{\sdot\lqqq} \newcommand{\ged}{\sdot\ge} \newcommand{\gqd}{\sdot\gq} \newcommand{\gqqd}{\sdot\gqq} \newcommand{\gqqqd}{\sdot\gqqq} \newcommand{\ld}{\sdot<} \newcommand{\ltd}{\sdot\lt} \newcommand{\lttd}{\sdot\ltt} \newcommand{\ltttd}{\sdot\lttt} \newcommand{\gd}{\sdot>} \newcommand{\gtd}{\sdot\gt} \newcommand{\gttd}{\sdot\gtt} \newcommand{\gtttd}{\sdot\gttt} % .. log versions (is it equal up to log?) \newcommand{\elog}{\slog=} \newcommand{\eqlog}{\slog\eq} \newcommand{\eqqlog}{\slog\eqq} \newcommand{\eqqqlog}{\slog\eqqq} \newcommand{\lelog}{\slog\le} \newcommand{\lqlog}{\slog\lq} \newcommand{\lqqlog}{\slog\lqq} \newcommand{\lqqqlog}{\slog\lqqq} \newcommand{\gelog}{\slog\ge} \newcommand{\gqlog}{\slog\gq} \newcommand{\gqqlog}{\slog\gqq} \newcommand{\gqqqlog}{\slog\gqqq} \newcommand{\llog}{\slog<} \newcommand{\ltlog}{\slog\lt} \newcommand{\lttlog}{\slog\ltt} \newcommand{\ltttlog}{\slog\lttt} \newcommand{\glog}{\slog>} \newcommand{\gtlog}{\slog\gt} \newcommand{\gttlog}{\slog\gtt} \newcommand{\gtttlog}{\slog\gttt} % % Miscellaneous \newcommand{\LHS}{\mathrm{LHS}} \newcommand{\RHS}{\mathrm{RHS}} % .. operators \DeclareMathOperator{\poly}{poly} \DeclareMathOperator{\polylog}{polylog} \DeclareMathOperator{\quasipoly}{quasipoly} \DeclareMathOperator{\negl}{negl} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\argmax}{arg\,max} % .. functions \DeclareMathOperator{\id}{id} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\err}{err} \DeclareMathOperator{\ReLU}{ReLU} % .. analysis \let\d\undefined \newcommand{\d}{\operatorname{d}\mathopen{}} \newcommand{\df}[2]{\f{\d #1}{\d #2}} \newcommand{\ds}[2]{\s{\d #1}{\d #2}} \newcommand{\part}{\partial} \newcommand{\partf}[2]{\f{\part #1}{\part #2}} \newcommand{\parts}[2]{\s{\part #1}{\part #2}} \newcommand{\grad}[1]{\mathop{\nabla\!_{#1}}} % .. sets \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathbb{F}} \newcommand{\zo}{\set{0,1}} \newcommand{\pmo}{\set{\pm 1}} % %%% SPECIALIZED MATH %%% % % Logic \renewcommand{\and}{\wedge} \newcommand{\AND}{\bigwedge} \newcommand{\or}{\vee} \newcommand{\OR}{\bigvee} \newcommand{\xor}{\oplus} \newcommand{\XOR}{\bigoplus} \newcommand{\union}{\cup} \newcommand{\inter}{\cap} \newcommand{\UNION}{\bigcup} \newcommand{\INTER}{\bigcap} \newcommand{\comp}{\overline} \newcommand{\true}{\r{true}} \newcommand{\false}{\r{false}} \newcommand{\tf}{\set{\true,\false}} \DeclareMathOperator{\One}{\mathbb{1}} \DeclareMathOperator{\1}{\mathbb{1}} % % Linear algebra \renewcommand{\span}{\mathrm{span}} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\Img}{Im} \newcommand{\transp}{\mathsf{T}} \renewcommand{\t}{^\transp} % ... named tensors \newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut \newcommand{\name}[1]{\mathsf{\namedtensorstrut #1}} \newcommand{\nbin}[2]{\mathbin{\underset{\substack{#1}}{\namedtensorstrut #2}}} \newcommand{\ndot}[1]{\nbin{#1}{\odot}} \newcommand{\ncat}[1]{\nbin{#1}{\oplus}} \newcommand{\nsum}[1]{\sum\limits_{\substack{#1}}} \newcommand{\nfun}[2]{\mathop{\underset{\substack{#1}}{\namedtensorstrut\mathrm{#2}}}} \newcommand{\ndef}[2]{\newcommand{#1}{\name{#2}}} \newcommand{\nt}[1]{^{\transp(#1)}} % % Probability \newcommand{\Normal}{\mathcal{N}} \let\Pr\undefined \DeclareMathOperator*{\Pr}{Pr} \DeclareMathOperator*{\G}{\mathbb{G}} \DeclareMathOperator*{\Odds}{Od} \DeclareMathOperator*{\E}{E} \DeclareMathOperator*{\Var}{Var} \DeclareMathOperator*{\Cov}{Cov} \DeclareMathOperator*{\corr}{corr} \DeclareMathOperator*{\median}{median} \newcommand{\dTV}{d_{\mathrm{TV}}} \newcommand{\dHel}{d_{\mathrm{Hel}}} \newcommand{\dJS}{d_{\mathrm{JS}}} % ... information theory \let\H\undefined \DeclareMathOperator*{\H}{H} \DeclareMathOperator*{\I}{I} \DeclareMathOperator*{\D}{D} % %%% SPECIALIZED COMPUTER SCIENCE %%% % % Complexity classes % .. classical \newcommand{\Poly}{\mathsf{P}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\PH}{\mathsf{PH}} \newcommand{\PSPACE}{\mathsf{PSPACE}} \renewcommand{\L}{\mathsf{L}} % .. probabilistic \newcommand{\formost}{\mathsf{Я}} \newcommand{\RP}{\mathsf{RP}} \newcommand{\BPP}{\mathsf{BPP}} \newcommand{\MA}{\mathsf{MA}} \newcommand{\AM}{\mathsf{AM}} \newcommand{\IP}{\mathsf{IP}} \newcommand{\RL}{\mathsf{RL}} % .. circuits \newcommand{\NC}{\mathsf{NC}} \newcommand{\AC}{\mathsf{AC}} \newcommand{\ACC}{\mathsf{ACC}} \newcommand{\TC}{\mathsf{TC}} \newcommand{\Ppoly}{\mathsf{P}/\poly} \newcommand{\Lpoly}{\mathsf{L}/\poly} % .. resources \newcommand{\TIME}{\mathsf{TIME}} \newcommand{\SPACE}{\mathsf{SPACE}} \newcommand{\TISP}{\mathsf{TISP}} \newcommand{\SIZE}{\mathsf{SIZE}} % .. keywords \newcommand{\co}{\mathsf{co}} \newcommand{\Prom}{\mathsf{Promise}} % % Boolean analysis \newcommand{\harpoon}{\!\upharpoonright\!} \newcommand{\rr}[2]{#1\harpoon_{#2}} \newcommand{\Fou}[1]{\widehat{#1}} \DeclareMathOperator{\Ind}{\mathrm{Ind}} \DeclareMathOperator{\Inf}{\mathrm{Inf}} \DeclareMathOperator{\Der}{\mathrm{D}} \DeclareMathOperator{\Stab}{\mathrm{Stab}} \DeclareMathOperator{\T}{T} \DeclareMathOperator{\sens}{\mathrm{s}} \DeclareMathOperator{\bsens}{\mathrm{bs}} \DeclareMathOperator{\fbsens}{\mathrm{fbs}} \DeclareMathOperator{\Cert}{\mathrm{C}} \DeclareMathOperator{\DT}{\mathrm{DT}} \DeclareMathOperator{\CDT}{\mathrm{CDT}} % canonical \DeclareMathOperator{\ECDT}{\mathrm{ECDT}} \DeclareMathOperator{\CDTv}{\mathrm{CDT_{vars}}} \DeclareMathOperator{\ECDTv}{\mathrm{ECDT_{vars}}} \DeclareMathOperator{\CDTt}{\mathrm{CDT_{terms}}} \DeclareMathOperator{\ECDTt}{\mathrm{ECDT_{terms}}} \DeclareMathOperator{\CDTw}{\mathrm{CDT_{weighted}}} \DeclareMathOperator{\ECDTw}{\mathrm{ECDT_{weighted}}} \DeclareMathOperator{\AvgDT}{\mathrm{AvgDT}} \DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree \DeclareMathOperator{\DTsize}{\mathrm{DT_{size}}} \DeclareMathOperator{\W}{\mathbf{W}} % .. functions (small caps sadly doesn't work) \DeclareMathOperator{\Par}{\mathrm{Par}} \DeclareMathOperator{\Maj}{\mathrm{Maj}} \DeclareMathOperator{\HW}{\mathrm{HW}} \DeclareMathOperator{\Th}{\mathrm{Th}} \DeclareMathOperator{\Tribes}{\mathrm{Tribes}} \DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}} \DeclareMathOperator{\CycleRun}{\mathrm{CycleRun}} \DeclareMathOperator{\SAT}{\mathrm{SAT}} \DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}} % % Dynamic optimality \newcommand{\OPT}{\mathsf{OPT}} \newcommand{\Alt}{\mathsf{Alt}} \newcommand{\Funnel}{\mathsf{Funnel}} % % Alignment \DeclareMathOperator{\Amp}{\mathrm{Amp}} % %%% TYPESETTING %%% % % In text \renewcommand{\th}{^{\mathrm{th}}} \newcommand{\degree}{^\circ} % % Fonts % .. bold \newcommand{\BA}{\boldsymbol{A}} \newcommand{\BB}{\boldsymbol{B}} \newcommand{\BC}{\boldsymbol{C}} \newcommand{\BD}{\boldsymbol{D}} \newcommand{\BE}{\boldsymbol{E}} \newcommand{\BF}{\boldsymbol{F}} \newcommand{\BG}{\boldsymbol{G}} \newcommand{\BH}{\boldsymbol{H}} \newcommand{\BI}{\boldsymbol{I}} \newcommand{\BJ}{\boldsymbol{J}} \newcommand{\BK}{\boldsymbol{K}} \newcommand{\BL}{\boldsymbol{L}} \newcommand{\BM}{\boldsymbol{M}} \newcommand{\BN}{\boldsymbol{N}} \newcommand{\BO}{\boldsymbol{O}} \newcommand{\BP}{\boldsymbol{P}} \newcommand{\BQ}{\boldsymbol{Q}} \newcommand{\BR}{\boldsymbol{R}} \newcommand{\BS}{\boldsymbol{S}} \newcommand{\BT}{\boldsymbol{T}} \newcommand{\BU}{\boldsymbol{U}} \newcommand{\BV}{\boldsymbol{V}} \newcommand{\BW}{\boldsymbol{W}} \newcommand{\BX}{\boldsymbol{X}} \newcommand{\BY}{\boldsymbol{Y}} \newcommand{\BZ}{\boldsymbol{Z}} \newcommand{\Ba}{\boldsymbol{a}} \newcommand{\Bb}{\boldsymbol{b}} \newcommand{\Bc}{\boldsymbol{c}} \newcommand{\Bd}{\boldsymbol{d}} \newcommand{\Be}{\boldsymbol{e}} \newcommand{\Bf}{\boldsymbol{f}} \newcommand{\Bg}{\boldsymbol{g}} \newcommand{\Bh}{\boldsymbol{h}} \newcommand{\Bi}{\boldsymbol{i}} \newcommand{\Bj}{\boldsymbol{j}} \newcommand{\Bk}{\boldsymbol{k}} \newcommand{\Bp}{\boldsymbol{p}} \newcommand{\Bq}{\boldsymbol{q}} \newcommand{\Br}{\boldsymbol{r}} \newcommand{\Bs}{\boldsymbol{s}} \newcommand{\Bt}{\boldsymbol{t}} \newcommand{\Bu}{\boldsymbol{u}} \newcommand{\Bv}{\boldsymbol{v}} \newcommand{\Bw}{\boldsymbol{w}} \newcommand{\Bx}{\boldsymbol{x}} \newcommand{\By}{\boldsymbol{y}} \newcommand{\Bz}{\boldsymbol{z}} \newcommand{\Balpha}{\boldsymbol{\alpha}} \newcommand{\Bbeta}{\boldsymbol{\beta}} \newcommand{\Bgamma}{\boldsymbol{\gamma}} \newcommand{\Bdelta}{\boldsymbol{\delta}} \newcommand{\Beps}{\boldsymbol{\eps}} \newcommand{\Bveps}{\boldsymbol{\veps}} \newcommand{\Bzeta}{\boldsymbol{\zeta}} \newcommand{\Beta}{\boldsymbol{\eta}} \newcommand{\Btheta}{\boldsymbol{\theta}} \newcommand{\Biota}{\boldsymbol{\iota}} \newcommand{\Bkappa}{\boldsymbol{\kappa}} \newcommand{\Blambda}{\boldsymbol{\lambda}} \newcommand{\Bmu}{\boldsymbol{\mu}} \newcommand{\Bnu}{\boldsymbol{\nu}} \newcommand{\Bxi}{\boldsymbol{\xi}} \newcommand{\Bomicron}{\boldsymbol{\omicron}} \newcommand{\Bpi}{\boldsymbol{\pi}} \newcommand{\Brho}{\boldsymbol{\rho}} \newcommand{\Bsigma}{\boldsymbol{\sigma}} \newcommand{\Btau}{\boldsymbol{\tau}} \newcommand{\Bupsilon}{\boldsymbol{\upsilon}} \newcommand{\Bphi}{\boldsymbol{\phi}} \newcommand{\Bfi}{\boldsymbol{\fi}} \newcommand{\Bchi}{\boldsymbol{\chi}} \newcommand{\Bpsi}{\boldsymbol{\psi}} \newcommand{\Bomega}{\boldsymbol{\omega}} % .. calligraphic \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CD}{\mathcal{D}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}} \newcommand{\CG}{\mathcal{G}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CI}{\mathcal{I}} \newcommand{\CJ}{\mathcal{J}} \newcommand{\CK}{\mathcal{K}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CN}{\mathcal{N}} \newcommand{\CO}{\mathcal{O}} \newcommand{\CP}{\mathcal{P}} \newcommand{\CQ}{\mathcal{Q}} \newcommand{\CR}{\mathcal{R}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CT}{\mathcal{T}} \newcommand{\CU}{\mathcal{U}} \newcommand{\CV}{\mathcal{V}} \newcommand{\CW}{\mathcal{W}} \newcommand{\CX}{\mathcal{X}} \newcommand{\CY}{\mathcal{Y}} \newcommand{\CZ}{\mathcal{Z}} % .. typewriter \newcommand{\TA}{\mathtt{A}} \newcommand{\TB}{\mathtt{B}} \newcommand{\TC}{\mathtt{C}} \newcommand{\TD}{\mathtt{D}} \newcommand{\TE}{\mathtt{E}} \newcommand{\TF}{\mathtt{F}} \newcommand{\TG}{\mathtt{G}} \newcommand{\TH}{\mathtt{H}} \newcommand{\TI}{\mathtt{I}} \newcommand{\TJ}{\mathtt{J}} \newcommand{\TK}{\mathtt{K}} \newcommand{\TL}{\mathtt{L}} \newcommand{\TM}{\mathtt{M}} \newcommand{\TN}{\mathtt{N}} \newcommand{\TO}{\mathtt{O}} \newcommand{\TP}{\mathtt{P}} \newcommand{\TQ}{\mathtt{Q}} \newcommand{\TR}{\mathtt{R}} \newcommand{\TS}{\mathtt{S}} \newcommand{\TT}{\mathtt{T}} \newcommand{\TU}{\mathtt{U}} \newcommand{\TV}{\mathtt{V}} \newcommand{\TW}{\mathtt{W}} \newcommand{\TX}{\mathtt{X}} \newcommand{\TY}{\mathtt{Y}} \newcommand{\TZ}{\mathtt{Z}}$

Adapted from this 3b1b video.

The dot product $u\cdot v \ce u_1v_1+\cdots+u_nv_n$

• is bilinear,
• gives a norm $\norm{u}^2_2 = \inner{u,u}$,
• is related to orthogonal projection.

See also: Geometric properties of linear algebra

Why in the world is it related to orthogonal projection?

Claim. If $u$ is a unit vector, then $u\cdot v$ is the (signed) length of the projection of $v$ on the line containing $u$.

Proof. This projection operation is linear in $v$, so it’s enough to prove that the projection of each basis vector $e_i$ is indeed $u \cdot e_i =u_i$. And since $u$ and $e_i$ are both unit vectors, by symmetry we can equivalently project $u$ onto $e_i$, which amounts to taking its $i\th$ coordinate $u_i$.

General expression

More generally, let’s define the dot product $u \cdot v$ to be the product of the length of $u$ and of $v$’s orthogonal projection onto $u$ (with a minus sign if they point in opposite directions), and try to figure out how to compute it.

We can geometrically verify that it is symmetric

and linear in the second component

so it’s a bilinear operation.

This means that we can decompose it as

\[ u \cdot v = \sum_{i,j} u_iv_i (e_i \cdot e_j) \]

where $e_i$ is the $i\th$ basis vector. And it’s clear that $e_i \cdot e_j$ should be $1$ iff $i=j$ and $0$ otherwise, so we get

\[ u \cdot v = u_1v_1 + \cdots+ u_n v_n. \]

TODO:

• I’m still somehow deeply unsatisfied with both these proofs.
• Why does the dot product not change if your rotate both vectors? It must be because the transpose of a rotation matrix is its own inverse, but why is that the case?