From 555464f3cbac5ad7fdc263c1b4b4a8ed3191c3f3 Mon Sep 17 00:00:00 2001 From: Mars Ultor Date: Sun, 21 Sep 2025 18:16:48 -0500 Subject: [PATCH] all done --- .gitignore | 2 + general-template/preamble.tex | 8 ++ labs-set-1/preamble.tex | 6 + labs-set-1/template.pdf | Bin 168359 -> 264816 bytes labs-set-1/template.tex | 212 ++++++++++++++++++++++++++++++++++ labs-set-1/template.toc | 4 + 6 files changed, 232 insertions(+) diff --git a/.gitignore b/.gitignore index 0a2d00d..5251fd9 100644 --- a/.gitignore +++ b/.gitignore @@ -7,3 +7,5 @@ **/*.bcf **/*.run.xml **/*.bbl +**/**.dat +**/**.script diff --git a/general-template/preamble.tex b/general-template/preamble.tex index bc65870..9f070cd 100644 --- a/general-template/preamble.tex +++ b/general-template/preamble.tex @@ -33,6 +33,12 @@ \usepackage{tikz} \usepackage{titletoc} +% add this to the preamble (near your other \usepackage lines) +\usepackage{pgfplots} +\pgfplotsset{compat=1.18} % or compat=newest if you prefer +\usepgfplotslibrary{colormaps} % enables 'viridis' etc. + + \newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} \SetCommentSty{mycommfont} \newcommand{\incfig}[1]{% @@ -40,6 +46,8 @@ \import{./figures/}{#1.pdf_tex} } + + \usepackage{tikzsymbols} \tikzset{ symbol/.style={ diff --git a/labs-set-1/preamble.tex b/labs-set-1/preamble.tex index 5ba904f..9c37981 100644 --- a/labs-set-1/preamble.tex +++ b/labs-set-1/preamble.tex @@ -32,6 +32,12 @@ \usepackage{tikz} \usepackage{titletoc} +% add this to the preamble (near your other \usepackage lines) +\usepackage{pgfplots} +\pgfplotsset{compat=1.18} % or compat=newest if you prefer +\usepgfplotslibrary{colormaps} % enables 'viridis' etc. + + \newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} \SetCommentSty{mycommfont} \newcommand{\incfig}[1]{% diff --git a/labs-set-1/template.pdf b/labs-set-1/template.pdf index 5b119066efcc03428c17e8532533a75b363a6814..c04ee06a7f219963fd2eff0de7a9a6696e270411 100644 GIT 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z=r>k!p7EAoeWqogpBS@vcjdNEZnOWevu~QFHH|AZFQvGoC^az`!~$)RFD^+eDyb++ UP2(~&GdD3a^a}0A9YN!~g&Q diff --git a/labs-set-1/template.tex b/labs-set-1/template.tex index 4052c60..7fdb26e 100644 --- a/labs-set-1/template.tex +++ b/labs-set-1/template.tex @@ -331,4 +331,216 @@ \end{align*} } +\qs{}{ + In order to aptly encode the direction and magnitude of the side of the triangle let $\vec{s}_1, \vec{s}_2$ originate from $P$. + \begin{align*} + \vec{s}_1 = \langle 6-1,6-1,-3-1 \rangle = \langle 5,5,-4 \rangle \quad \vec{s}_2 = \langle 10-1,4-1,2-1 \rangle =\langle 9,3,1 \rangle \\ + A=\tfrac{1}{2}\|\vec{s}_1\times\vec{s}_2\| = \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + 5 & 5 & -4 \\ + 9 & 3 & 1 + \end{vmatrix} \\ + \implies \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + 5 & 5 & -4 \\ + 9 & 3 & 1 + \end{vmatrix} = \hat{i}(5+12)-\hat{j}(5+36)+\hat{k}(15-45)=\langle 17,-41,-30\rangle \\ + \implies \tfrac{1}{2}\|\vec{s}_1\times\vec{s}_2\| = \tfrac{1}{2}\cdot\sqrt{17^2+41^2+30^2} = \tfrac{1}{2}\sqrt{2870} \\ + \text{This corresponds with answer choice D} + \end{align*} +} + +\qs{}{ + \begin{align*} + \|\vec{F}\times\vec{PQ}\| =\|\vec{\tau}\| = \|\vec{PQ}\|\cdot\|\vec{F}\|\sin{45\degree}=\tfrac{9}{12}\cdot5\frac{\sqrt{2}}{2}=\frac{45\sqrt{2}}{24} = \tfrac{15}{8}\sqrt{2}\;\mathrm{ft}\cdot\mathrm{lb} \\ + \text{This corresponds with answer choice D} + \end{align*} +} + +\chapter{Lab 5 - 13.5} +\section{Work} + +\qs{}{ + \begin{align*} + \vec{r}=\langle 0,1,0 \rangle + \langle 3,0,-1 \rangle t \\ + \text{breaking it down into component form:}\\ + x=3t,\quad y=1,\quad z=-t \\ + \text{This corresponds with answer choice C} + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{r}=\langle -6,5,-5 \rangle t + \langle 5,-1,-5 \rangle \\ + \text{This corresponds with answer choice C} + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{v}=P_2-P_1=\langle 3,0,4 \rangle \quad \vec{r}=\langle 3,0,4 \rangle t + \langle -3,7,3 \rangle \lor\vec{r} = \langle 3,0,4 \rangle t + \langle 0,7,7 \rangle \\ + \text{This corresponds with answer choice B} + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{u}=\langle 6,4,4 \rangle \quad \vec{v}= \langle -7,-6,-4 \rangle \\ + \vec{v}_1\coloneqq \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + 6 &4 & 4 \\ + -7 & -6 & -4 +\end{vmatrix} = \hat{i}(-16+24)-\hat{j}(-24+28)+\hat{k}(-36+28)=\langle 8,-4,-8\rangle \\ +\vec{r}=\vec{v}_1 t+\langle -4,-7,4 \rangle \\ +\text{This corresponds with answer choice D} + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{v}_1=\langle 3,-3,-1 \rangle \quad \vec{v}_2 = \langle 4,-2,-5 \rangle \\ \nexists k | k\in \mathbb{R} \land k\cdot\vec{v}_1=\vec{v}_2\\ \therefore l_1 \nparallel l_2 \\ +1+3t &= 3+4s \\ +3-3t &= -1-2s \\ +z=-t &= 3-5s \\[6pt] +s &= \tfrac{1}{4}(3t-2) \\ +3-3t &= -1-2\cdot\tfrac{1}{4}(3t-2) \\ +3-3t &= -1-\tfrac{1}{2}(3t-2) \\ +3-3t &= -\tfrac{3}{2}t \\ +3 &= \tfrac{3}{2}t \\ +t &= 2 \\ +s &= \tfrac{1}{4}(3\cdot 2-2)=1 \\[6pt] +x &= 1+3(2)=7 \\ +y &= 3-3(2)=-3 \\ +z &= -2 \\[6pt] +x &= 3+4(1)=7 \\ +y &= -1-2(1)=-3 \\ +z &= 3-5(1)=-2 \\[6pt] +\therefore (x,y,z) &= (7,-3,-2) \\ +\text{This corresponds with answer choice D} +\end{align*} + +} +\qs{}{ + \begin{align*} + \exists k | k\in \mathbb{R} \land k\cdot\vec{v}_1=\vec{v}_2\\ \therefore l_1 \parallel l_2 \\ + 3-t&=-6+5s \\ + 9-t&=5s \\ + 9-5s&=t \\ + 1+6(9-5s)&=55-30s=1+54-30s \\ + 55-30s&=55-30s \forall s\in\mathbb{R} \\ + 4-2(9-5s)&=-14+10s=4-18+10s \\ + -14+10s&=-14+10s \forall s\in\mathbb{R} \\ + \therefore \text{$l_1$ and $l_2$ are identical. This corresponds to answer choice B} + \end{align*} +} + +\qs{}{ + \begin{align*} + \vec{n}\cdot(\vec{r}-\vec{r}_0)=0 \quad \langle 2,7,6 \rangle \cdot(\vec{r}-\langle 4,-3,2 \rangle) = 0 \\ + \vec{r}|\vec{r}=\langle x,y,z \rangle \implies \langle 2,7,6 \rangle \cdot \langle x-4,y+3,z-2 \rangle \\ + 2(x-4)+7(y+3)+6(z-2)=2x-8+7y+21+6z-12=0 \\ + 2x+7y+6z-8+21-12=0 \quad 2x+7y+6z=-1 \\ + \text{This corresponds with answer choice C} + \end{align*} +} + +\qs{}{ + \begin{align*} + \text{Considering answer choice A: } \\ + 5(-1)-7(8)+58=-5-56+58=-3\ne 3 \therefore \cancel{\mathrm{A}} \\ + 5(-1)+7(8)-58=-5+56-58=51-58=-7\ne3\therefore\cancel{\mathrm{B}} \\ + 5(-1)+y(8)-58\ne-3 \therefore \cancel{\mathrm{C}} \\ + 5(-1)-7(8)+58=-5-56+58=-3 \; \therefore \; \text{The answer is D} + \end{align*} +} + +\qs{}{ + \begin{align*} + -2x+2&=-2y \\ + -2x=2+5z&=2 \\ + z_0\coloneqq0 \quad -2y+5(0)=2 \; \implies y=-2 \\ + -2x+2(-2)&=-2=-2x-4 \\ + -2x&=2 \\ + \implies x&=-1 \\ + r_0=(-1,-2,0) \\ + \text{By parameterizing $r_0$, only answer choice D has the necessary constant terms.} + \end{align*} +} + +\qs{}{ + \begin{align*} + x+y&=7-z \\ + 7-z&=12 \\ + z&=-5 \\ + x&=t \\ + t+y&=12 \\ + y&=12-t \\ + \therefore x=t \quad y=12-t \quad z=-5 + \end{align*} + This corresponds with answer choice B. Answer Choice A is the same line with different parametriazation to $x=-t$ instead of $x=t$ +} + +\chapter{Lab 6 - 13.6} +\section{Work} + +\qs{}{ + \begin{tikzpicture} + \begin{axis}[ + legend pos=outer north east, + title=Q1, + axis lines = box, + xlabel = $x$, + ylabel = $y$, + zlabel = $z$, + view={45}{30}, + ] + \addplot3[ + mesh, + samples=50, + color=blue, + ] + {sqrt(10*x - 2*y^2/5)}; +\addplot3[ + mesh, + samples=50, + color=red, +] +{-sqrt(10*x - 2*y^2/5)}; + \addlegendentry{$z = \pm\sqrt{10x - \frac{2y^2}{5}}$} + \end{axis} + \end{tikzpicture} + + This image corresponds to that of an elliptical paraboloid, corresponding with answer choice A. +} + +\qs{}{ + Since all terms are raised to the second degree, all behave linearly with respect to each other. Therefore, this is not a curved surface. Furthermore, the coefficient of the term containing $x$ is negative. Therefore, the equation represents an elliptical cone along the $x$-axis. This corresponds with answer choice D. +} + +\qs{}{ + This surface is linear along the $y$-axis, and has a negative coefficient on the $x$ term. Therefore, the surface is a Hyperbolic paraboloid, corresponding with answer choice A. +} +\newpage +\qs{}{ + The surface is linear along the $z$-axis, guaranteeing it to be a paraboloid. The $xz$-trace occurs when $y=0$. The $yz$-trace occurs when $x=0$. + \begin{align*} + \text{$xz$-trace: }\quad 16x^2-16(0)^2=z=16x^2 \\ + \text{$yz$-trace: }\quad 16(0)^2-16y^2-z=0=16y^2+z + \end{align*} + This corresponds with answer choice D. +} + +\qs{}{ + Since two terms have a negative coefficient, and all terms are quadratic, the surface must be a hyperboloid of two sheets, leaving only answer choices B and D. A hyperboloid of two sheets only has hyperbolic cross-sections, so B must be the answer. +} + +\qs{}{ + Since the equation only yields circular cross sections, and varies quadratically over $\hat{i},\hat{j},\hat{k}$, The answer is Figure 1, A. +} + +\qs{}{ + The equation yield a circular cross section for any plane parallel to the $yz$-plane, and only discontinuous sets for the $xy$-plane and the $xz$-plane, the surface must be elliptical $\forall (y,z)$. Only Figure 3 satisfies the conditions, validating answer choice C. +} + + \end{document} diff --git a/labs-set-1/template.toc b/labs-set-1/template.toc index d42b737..10736c8 100644 --- a/labs-set-1/template.toc +++ b/labs-set-1/template.toc @@ -4,4 +4,8 @@ \contentsline {section}{\numberline {2.1}Work}{5}{section.2.1}% \contentsline {chapter}{\numberline {3}Lab 4 - 13.3, 13.4}{7}{chapter.3}% \contentsline {section}{\numberline {3.1}Work}{7}{section.3.1}% +\contentsline {chapter}{\numberline {4}Lab 5 - 13.5}{11}{chapter.4}% +\contentsline {section}{\numberline {4.1}Work}{11}{section.4.1}% +\contentsline {chapter}{\numberline {5}Lab 6 - 13.6}{14}{chapter.5}% +\contentsline {section}{\numberline {5.1}Work}{14}{section.5.1}% \contentsfinish