Dot Products and Projections

Illegal control sequence name for \newcommand $\newenvironment {prompt}{}{} \newcommand {\DeclareMathOperator }[2]{\@OldDeclareMathOperator {##1}{##2}\immediate \write \myfile {\unexpanded {\DeclareMathOperator }{\unexpanded {##1}}{\unexpanded {##2}}}} \newcommand {\reserved@a }[2]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{} \newcommand {\AppendGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Append \grfext@Check }{\grfext@Append \grfext@@Add }} \newcommand {\PrependGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Prepend \grfext@Check }{\grfext@Prepend \grfext@@Add }} \newcommand {\RemoveGraphicsExtensions }[1]{\@ifundefined {Gin@extensions}{\def \Gin@extensions {}}{\edef \grfext@tmp {\zap@space ##1 \@empty }\@for \grfext@ext :=\grfext@tmp \do {\def \grfext@next {\let \grfext@tmp \Gin@extensions \@expandtwoargs \@removeelement \grfext@ext \Gin@extensions \Gin@extensions \ifx \grfext@tmp \Gin@extensions \let \grfext@next \relax \fi \grfext@next }\grfext@next }}\grfext@Print \RemoveGraphicsExtensions } \newcommand {\epstopdfsetup }[0]{\setkeys {ETE}} \newcommand {\epstopdfcall }[1]{\ifETE@InsideSetfile \expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi {‘##1}{\Gin@base \Gin@ext }} \newcommand {\epstopdfDeclareGraphicsRule }[4]{\ifx \\##4\\\@PackageError {epstopdf-base}{Conversion command is missing}\@ehc \else \begingroup \@ifundefined {Gin@rule@##1}{}{\@PackageInfo {epstopdf-base}{Redefining graphics rule for ‘##1’}}\endgroup \@namedef {Gin@rule@##1}####1{{##2}{##3}{\epstopdfcall {##4}}}\fi } \newcommand {\GetTitleStringSetup }[0]{\setkeys {gettitlestring}} \newcommand {\GetTitleString }[0]{\ifGTS@expand \expandafter \GetTitleStringExpand \else \expandafter \GetTitleStringNonExpand \fi } \newcommand {\GetTitleStringExpand }[1]{\def \GetTitleStringResult {##1}\begingroup \GTS@DisablePredefinedCmds \GTS@DisableHook \edef \x {\endgroup \noexpand \def \noexpand \GetTitleStringResult {\GetTitleStringResult }}\x } \newcommand {\GetTitleStringNonExpand }[1]{\def \GetTitleStringResult {##1}\global \let \GTS@GlobalString \GetTitleStringResult \begingroup \GTS@RemoveLeft \GTS@RemoveRight \endgroup \let \GetTitleStringResult \GTS@GlobalString } \newcommand {\GTS@DisableHook }[0]{} \newcommand {\label@hook }[0]{} \newcommand {\@currentlabelname }[0]{} \newcommand {\reserved@a }[0]{} \newcommand {\@safe@activestrue }[0]{} \newcommand {\@safe@activesfalse }[0]{} \newcommand {\nameref }[0]{\@ifstar \T@nameref \T@nameref } \newcommand {\Sectionformat }[2]{##1} \newcommand {\vnameref }[1]{\unskip ~\nameref {##1}\@vpageref [\unskip ]{##1}} \newcommand {\ref }[0]{\@ifstar \@refstar \T@ref } \newcommand {\pageref }[0]{\@ifstar \@pagerefstar \T@pageref } \newcommand {\nameref }[0]{\@ifstar \@namerefstar \T@nameref } \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{}$

Objectives:

1.: Know how to compute a dot product.
2.: Know the cosine formula for the dot product.
3.: Understand what the dot product formula says about the angle between vectors.
4.: Know what the projection looks like geometrically and how to compute it.

Recap Video

You can watch watch the following video which recaps the ideas of the section.

Test your understanding with the following questions.

If $\textbf {v} = \left \langle v_1,v_2,v_3 \right \rangle$ and $\textbf {w} = \left \langle w_1,w_2,w_3 \right \rangle$ , then the dot product $\textbf {v} \cdot \textbf {w}$ is equal to which of the following (select all that apply):

If $\textbf {v}$ and $\textbf {w}$ are two nonzero vectors, then which of the following is true of the projection $\text {proj}_{\textbf {w}} \textbf {v}$ ? Select all that apply.

Problems

If $\textbf {v} = \left \langle 1,2,1 \right \rangle$ and $\textbf {w} = \left \langle -2,-1,3 \right \rangle$ , then $\textbf {v} \cdot \textbf {w} = \answer {-1}$ .

The angle between the two vectors is .

What it the angle between the vectors $\textbf {v} = \left \langle 1,2,1 \right \rangle$ and $\textbf {w}=\left \langle 0,1,-3 \right \rangle$ ?

We will use the formula

$\textbf {v} \cdot \textbf {w} = \| \textbf {v} \| \| \textbf {w}\| \cos (\theta ).$

The dot product of $\textbf {v}$ and $\textbf {w}$ is $\textbf {v} \cdot \textbf {w} = \answer {-1}$ .
The length of $\textbf {v}$ is $\| \textbf {v} \| = \answer {\sqrt {6}}$ and the length of $\textbf {w}$ is $\|\textbf {w}\| = \answer {\sqrt {10}}$
The dot product formula says that $\cos (\theta ) = \frac {\textbf {v} \cdot \textbf {w}}{\|\textbf {v}\| \|\textbf {w}\|} = -\frac {\answer {1}}{\answer {\sqrt {60}}}$
Therefore, $\theta = \answer {\arccos \left (\frac {-1}{\sqrt {60}} \right )}$

If $\textbf {u}$ and $\textbf {v}$ are unit vectors with an angle of $\pi /3$ between them, then $\textbf {u} \cdot \textbf {v} = \answer {1/2}$ .

Use $\textbf {u} \cdot \textbf {v} = \| \textbf {u} \| \| \textbf {v} \| \cos (\theta )$ .

Problems

Let $\textbf {v} = \left \langle 1,2,0 \right \rangle$ an $\textbf {w} = \left \langle 3,5,1 \right \rangle$ .

The dot product of $\textbf {v}$ and $\textbf {w}$ is $\textbf {v} \cdot \textbf {w} = \answer {13}$ .
The angle between $\textbf {v}$ and $\textbf {w}$ is
.
The projection $\text {proj}_{\textbf {w}} \textbf {v}$ will be in the
direction as $\textbf {w}$ .
The component $\text {comp}_{\textbf {w}} \textbf {v} = \answer {\frac {13}{\sqrt {35}}}$ .
The projection $\text {proj}_{\textbf {w}} \textbf {v} = \left \langle \answer {\frac {39}{35}}, \answer {\frac {65}{35}},\answer {\frac {13}{35}} \right \rangle .$

Suppose $\textbf {v}$ and $\textbf {w}$ are nonzero three-dimensional vectors with $\textbf {v}$ orthogonal to $\textbf {w}$ . Then

$\text {proj}_{\textbf {w}} \textbf {v} = \left \langle \answer {0},\answer {0},\answer {0} \right \rangle .$

True/False

If $\textbf {v}$ is a unit vector, then it must be true that $\textbf {v} \cdot \textbf {v} = 1$ .

Assume $\textbf {v}$ and $\textbf {w}$ are nonzero vectors with $\textbf {v} \cdot \textbf {w} = 0$ . Then it must be true that

$\| \textbf {v} + \textbf {w} \|^2 = \| \textbf {v} \|^2+ \| \textbf {w} \|^2.$

Assume $\textbf {v}$ and $\textbf {w}$ are nonzero vectors. Then it must be true that

$\text {proj}_{2\textbf {w}} \textbf {v} = 2\text {proj}_{\textbf {w}}\textbf {v}.$

If $\textbf {v}$ , $\textbf {w}$ , $\textbf {u}$ are nonzero vectors with $\textbf {v} \cdot \textbf {w} = 0$ and $\textbf {w} \cdot \textbf {u} = 0$ , then it must be true that $\textbf {v} \cdot \textbf {u}$ is always zero.

Optional: Dot Product Cosine Formula

For those who are interested, below is a video explaining where the dot product cosine formula comes from.

Optional: Projection Formula Explained

Below is a video explaining how to arrive at the projection formula.

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Recap Video

Problems

Problems

True/False

Optional: Dot Product Cosine Formula

Optional: Projection Formula Explained

Controls

Symbols

Settings